Graphing and Writing Linear Equations

Transcription

1 Graphing and Writing Linear Equations. Graphing Linear Equations. Slope of a Line. Graphing Linear Equations in Slope-Intercept Form. Graphing Linear Equations in Standard Form. Writing Equations in Slope-Intercept Form. Writing Equations in Point-Slope Form.7 Solving Real-Life Problems Oka Descartes, stand on the - ais and tr to intercept the pass when I throw. Copright Big Ideas Learning, LLC

2 What You Learned Before I estimate that we a slope of a bout -0. re on a ou think?. What do Eample Evaluate + ( + ) when = and = ( + ) = ()(7) + ( + 7) Substitute for and 7 for. = 8(7) + ( + 7) Use order of operations. = + () Simplif. = + Multipl. = 89 Add. Evaluate the epression when a = and b =.. a b. 8ab Eample b a Write the ordered pair that corresponds to Point U. Point U is units to the left of the origin and units down. So, the -coordinate is and the -coordinate is. The ordered pair (, ) corresponds to Point U. Eample. a + (b a ). Which point is located at (, )? Start at the origin. Move units right and units down. Q S N P R U T V Point T is located at (, ). Use the graph to answer the question.. Write the ordered pair that corresponds to Point Q.. Write the ordered pair that corresponds to Point P. 7. Which point is located at (, 0)? 8. Which point is located in Quadrant II? Copright Big Ideas Learning, LLC MSCC_ALG_PE_000_co.indd /9/ :: PM

3 . Graphing Linear Equations How can ou draw its graph? How can ou recognize a linear equation? ACTIVITY: Graphing a Linear Equation Work with a partner. a. Use the equation = + to complete the table. (Choose an two -values and find the -values.) b. Write the two ordered pairs given b the table. These are called solution points of the equation. c. PRECISION Plot the two solution points. Draw a line eactl through the two points. d. Find a different point on the line. Check that this point is a solution point of the equation = +. e. LOGIC Do ou think it is true that an point on the line is a solution = + Solution Points O point of the equation = +? Eplain. f. Choose five additional -values for the table. (Choose positive and negative -values.) Plot the five corresponding solution points. Does each point lie on the line? COMMON CORE Graphing Equations In this lesson, ou will understand that lines represent solutions of linear equations. graph linear equations. Learning Standards A.CED. A.REI.0 = + Solution Points g. LOGIC Do ou think it is true that an solution point of the equation = + is a point on the line? Eplain. h. THE MEANING OF A WORD Wh is = a + b called a linear equation? Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

4 ACTIVITY: Using a Graphing Calculator Math Practice Recognize Usefulness of Tools What are some advantages and disadvantages of using a graphing calculator to graph a linear equation? Use a graphing calculator to graph = +. a. Enter the equation = + into our calculator. b. Check the settings of the viewing window. The boundaries of the graph are set b the minimum and maimum - and -values. The number of units between the tick marks are set b the - and -scales. This is the standard viewing window. c. Graph = + on our calculator d. Change the settings of the viewing window to match those shown. 8 Compare the two graphs.. IN YOUR OWN WORDS How can ou recognize a linear equation? How can ou draw its graph? Write an equation that is linear. Write an equation that is not linear.. Use a graphing calculator to graph = in the standard viewing window. a. Can ou tell where the line crosses the -ais? Can ou tell where the line crosses the -ais? b. How can ou adjust the viewing window so that ou can determine where the line crosses the - and -aes?. CHOOSE TOOLS You want to graph =..8. Would ou graph it b hand or using a graphing calculator? Wh? Use what ou learned about graphing linear equations to complete Eercises and on page. Copright Big Ideas Learning, LLC Section. Graphing Linear Equations

5 . Lesson Lesson Tutorials Ke Vocabular linear equation, p. solution of a linear equation, p. Linear Equations A linear equation is an equation whose graph is a line. The points on the line are solutions of the equation. You can use a graph to show the solutions of a linear equation. The graph below is for the equation = +. Remember An ordered pair (, ) is used to locate a point in a coordinate plane. (, ) 0 (, 0) (, 0) 0 (0, ) (, ) (, ) = + (0, ) EXAMPLE Graphing a Linear Equation Graph = +. Step : Make a table of values. Check â Ź à Ź = + (, ) = ( ) + (, ) = (0) + (0, ) 0 = () + (, ) = + (0, ) Step : Plot the ordered pairs. Ź (, ) Step : Draw a line through the points. (, ) Graphing Horizontal and Vertical Lines The graph of = b is a horizontal line passing through (0, b). The graph of = a is a vertical line passing through (a, 0). âb (a, 0) (0, b) O Chapter MSCC_ALG_PE_00.indd âa O Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ ::0 PM

6 EXAMPLE Graphing a Horizontal Line and a Vertical Line a. Graph =. b. Graph =. The graph of = is a horizontal line passing through (0, ). Draw a horizontal line through this point. The graph of = is a vertical line passing through (, 0). Draw a vertical line through this point. O (0, ) (, 0) O Eercises Graph the linear equation. Use a graphing calculator to check our graph, if possible.. =. = +. =. =. EXAMPLE Real-Life Application The wind speed (in miles per hour) of a tropical storm is = +, where is the number of hours after the storm enters the Gulf of Meico. a. Graph the equation. b. When does the storm become a hurricane? a. Make a table of values. = + (, ) 0 = (0) + (0, ) 7 = + = () + 8 (, 8) 7 A tropical storm becomes a hurricane when wind speeds are at least 7 miles per hour. = () + 70 (, 70) = () + 7 (, 7) 8 Plot the ordered pairs and draw a line through the points. 0 b. From the graph, ou can see that = 7 when =. So, the storm becomes a hurricane hours after it enters the Gulf of Meico.. WHAT IF? In Eample, the wind speed of the storm is =. +. When does the storm become a hurricane? Copright Big Ideas Learning, LLC Section. Graphing Linear Equations

7 Eercises. Help with Homework. VOCABULARY What tpe of graph represents the solutions of the equation = +?. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. = = + = = + )= 9+(- )= +(- 9)= +(- = ) 9+(- PRECISION Cop and complete the table. Plot the two solution points and draw a line eactl through the two points. Find a different solution point on the line... = = + Graph the linear equation. Use a graphing calculator to check our graph, if possible.. = 9. =. = 7. =. = 8. = 0. = 7. = +. =. =.7. = 0.. = 7. ERROR ANALYSIS Describe and correct the error in graphing the equation. 8. MESSAGING You sign up for an unlimited tet messaging plan for our cell phone. The equation = 0 represents the cost (in dollars) for sending tet messages. Graph the equation. What does the graph tell ou? = O (, 0) 9. MAIL The equation = + represents the cost (in dollars) of mailing a package that weighs pounds. a. Graph the equation. b. Use the graph to estimate how much it costs to mail the package. c. Use the equation to find eactl how much it costs to mail the package. Chapter MSCC_ALG_PE_00.indd Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ ::09 PM

8 Solve for. Then graph the equation. Use a graphing calculator to check our graph. 0. =. + =. + = =.. SAVINGS You have $00 in our savings account and plan to deposit $.0 each month. a. Write and graph a linear equation that represents the balance in our account. b. How man months will it take ou to save enough mone to bu 0 acres of land on Mars? Video time: min. 0 sec.. CAMERA One second of video on our digital camera uses the same amount of memor as two pictures. Your camera can store 0 pictures. a. Write and graph a linear equation that represents the number of pictures our camera can store if ou take seconds of video. b. How man pictures can our camera store after ou take the video shown?. PROBLEM SOLVING Along the U.S. Atlantic Coast, the sea level is rising about millimeters per ear. How man millimeters has sea level risen since ou were born? How do ou know? Use a linear equation and a graph to justif our answer. 7. The sum S of the measures of the angles of a polgon is S = (n ) 80, where n is the number of sides of the polgon. a. Plot four points (n, S ) that satisf the equation. Do the points lie on a line? Eplain our reasoning. b. Does the value n =. make sense in the contet of the problem? Eplain our reasoning. Write the ordered pair corresponding to the point. (Skills Review Handbook) B 8. Point A 9. Point B A 0. Point C. Point D. MULTIPLE CHOICE A debate team has female members. The ratio of females to males is :. How man males are on the debate team? (Skills Review Handbook) O A B 0 C D D C Copright Big Ideas Learning, LLC Section. Graphing Linear Equations 7

9 . Slope of a Line describe the line? How can the slope of a line be used to Slope is the rate of change between an two points on a line. It is the measure of the steepness of the line. To find the slope of a line, find the ratio of the change in (vertical change) to the change in (horizontal change). change in slope = change in 7 Slope = 7 ACTIVITY: Finding the Slope of a Line Work with a partner. Find the slope of each line using two methods. Method : Use the two black points. Method : Use the two pink points. Do ou get the same slope using each method? Wh do ou think this happens? a. b. COMMON CORE Slope of a Line In this lesson, ou will find slopes of lines using two points. find slopes of lines from tables. Preparing for Standards F.IF. F.IF. c. d. 8 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

10 ACTIVITY: Drawing Lines with Given Slopes Math Practice Make Conjectures What does the slope tell ou about the graph of the line? Eplain. Work with a partner. Draw a line through the black point using the given slope. Draw a line through the pink point using the given slope. What do ou notice about the two lines? a. Slope = b. Slope = ACTIVITY: Drawing Lines with Given Slopes Work with a partner. Eamine the lines drawn through the black points in parts (a) and (b) of Activit. Draw these two lines in the same coordinate plane. Describe the angle formed b the two lines. What do ou notice about the product of the slopes of the two lines?. IN YOUR OWN WORDS How can the slope of a line be used to describe the line?. Based on our results in Activit, make a conjecture about two different nonvertical lines in the same plane that have the same slope.. REPEATED REASONING Repeat Activit for the lines drawn through the pink points in Activit. Based on our results, make a conjecture about two lines in the same plane whose slopes have a product of. Use what ou learned about the slope of a line to complete Eercises on page. Copright Big Ideas Learning, LLC Section. Slope of a Line 9

11 . Lesson Lesson Tutorials Ke Vocabular slope, p. 0 rise, p. 0 run, p. 0 Slope The slope of a line is a ratio of the change in (the rise) to the change in (the run) between an two points, (, ) and (, ), on the line. (, ) Run â Ź Negative slope Positive slope In the slope formula, is read as sub one and is read as sub two. The numbers and in and are called subscripts. O O The line rises from left to right. EXAMPLE Rise â Ź O rise change in slope = = = run change in Reading (, ) The line falls from left to right. Finding the Slope of a Line Describe the slope of the line. Then find the slope. a. (, ) The line rises from left to right. So, the slope is positive. Let (, ) = (, ) and (, ) = (, ). 0 Chapter MSCC_ALG_PE_00.indd 0 (, ) slope = When finding slope, ou can label either point as (, ) and the other point as (, ). (, ) Stud Tip (, ) b. The line falls from left to right. So, the slope is negative. Let (, ) = (, ) and (, ) = (, ). slope = ( ) ( ) = =, or = = Graphing and Writing Linear Equations ( ) Copright Big Ideas Learning, LLC /9/ ::9 PM

12 Find the slope of the line. Eercises (, ) (, ) Ź EXAMPLE. Ź (, ) (, ) (Ź, Ź) Ź (, ) Ź Finding the Slope of a Horizontal Line Find the slope of the line. There is no change in. So, the change in is 0. (, ) (, ) slope = ( ) = 0 7 =, or 0 The slope is 0. EXAMPLE Finding the Slope of a Vertical Line Find the slope of the line. There is no change in. So, the change in is 0. slope = 0 (, ) = = (, ) Ź Because division b zero is undefined, the slope of the line is undefined. Find the slope of the line through the given points. Eercises. (, ), (7, ). (, ), (, ). (0, 8), (0, 0) 7. How do ou know that the slope of ever horizontal line is 0? How do ou know that the slope of ever vertical line is undefined? Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd Section. Slope of a Line /9/ :: PM

13 EXAMPLE Finding Slope from a Table The points in the table lie on a line. How can ou find the slope of the line from the table? What is the slope? Choose an two points from the table and use the slope formula. Use the points (, ) = (, 8) and (, ) = (, ). slope = = 8 = The slope is. Check 8 7 (, 8) (, ) (7, ) (0, ) Eercises The points in the table lie on a line. How can ou find the slope of the line from the table? What is the slope? Slope Positive slope Negative slope Slope of 0 Undefined slope O O O O The line rises The line falls The line is The line is from left to right. from left to right. horizontal. vertical. Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

14 Eercises. Help with Homework. CRITICAL THINKING Refer to the graph. A B a. Which lines have positive slopes? b. Which line has the steepest slope? C c. Do an of the lines have undefined slope? Eplain.. OPEN-ENDED Describe a real-life situation in which ou need to know the slope.. REASONING The slope of a line is 0. What do ou know about the line? )= 9+(- )= +(- 9)= +(- = ) 9+(- Draw a line through each point using the given slope. What do ou notice about the two lines?. Slope =. Slope =. Slope = Find the slope of the line (, ) 9. (, ). (, ). (, ) Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd Ź Ź Ź (, ) (, ) (, ) (, ) (, 0) 0. (, 0) (, ) Ź (, Ź) Ź Section. Slope of a Line /9/ ::7 PM

15 Find the slope of the line through the given points.. (, ), (, ). (, ), (, 8). ( 7, 0), ( 7, ). (, ), (, ) 7. (0, ), (, ) 8. (, ), (, ) 9. ERROR ANALYSIS Describe and correct the error in finding the slope of the line. 0. CRITICAL THINKING Is it more difficult to walk up the ramp or the hill? Eplain. Slope = = = (, ) (, ) ft hill ramp 8 ft 8 ft ft The points in the table lie on a line. How can ou find the slope of the line from the table? What is the slope? ft ft. PITCH Carpenters refer to the slope of a roof as the pitch of the roof. Find the pitch of the roof.. PROJECT The guidelines for a wheelchair ramp suggest that the ratio of the rise to the run be no greater than :. a. CHOOSE TOOLS Find a wheelchair ramp in our school or neighborhood. Measure its slope. Does the ramp follow the guidelines? b. Design a wheelchair ramp that provides access to a building with a front door that is. feet higher than the sidewalk. Illustrate our design. Use an equation to find the value of k so that the line that passes through the given points has the given slope. 7. (, ), (, k); slope = 8. (, k), (, 0); slope = 9. (, k), (, 7); slope = 0. (, ), (k, ); slope = Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

16 . TURNPIKE TRAVEL The graph shows the cost of traveling b car on a turnpike. a. Find the slope of the line. b. Eplain the meaning of the slope as a rate of change. Cost (dollars).0.0 Turnpike Travel. BOAT RAMP Which is steeper: the boat ramp or a road with a % grade? Eplain. (Note: Road grade is the vertical increase divided b the horizontal distance.) Miles driven ft ft. REASONING Do the points A(, ), B(, ), and C(, ) lie on the same line? Without using a graph, how do ou know?. BUSINESS A small business earns a profit of $00 in Januar and $7,00 in Ma. What is the rate of change in profit for this time period?. STRUCTURE Choose two points in the coordinate plane. Use the slope formula to find the slope of the line that passes through the two points. Then find the slope using the formula. Are our results the same? Eplain.. The top and bottom of the slide are level with the ground, which has a slope of 0. a. What is the slope of the main portion of the slide? b. How does the slope change if the bottom of ft the slide is onl inches above the ground? Is the slide steeper? Eplain. 8 in. ft ft 8 ft Graph the linear equation. (Section.) 7. = 8. = 0. MULTIPLE CHOICE What is the prime factorization of 8? (Skills Review Handbook) 9. = A 7 B 7 C 7 D Copright Big Ideas Learning, LLC Section. Slope of a Line

17 Etension Slopes of Parallel and. Perpendicular Lines Lesson Tutorials Ke Vocabular perpendicular lines, p. 7 Parallel Lines and Slopes â à Two different lines in the same plane that never intersect are parallel lines. Nonvertical parallel lines have the same slope. Stud Tip Vertical lines have undefined slopes. Ź Ź Ź Ź All vertical lines are parallel. â Ź Identifing Parallel Lines EXAMPLE Which two lines are parallel? How do ou know? Find the slope of each line. Blue Line Red Line Green Line slope = slope = slope = 0 = =, or = =, or =, or ( ) = (, ) (, ) (, ) (, ) (0, ) (, ) The slope of the blue and green lines is. The slope of the red line is. The blue and green lines have the same slope, so the are parallel. Which lines are parallel? How do ou know?.. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Are the given lines parallel? Eplain our reasoning.. =, =. = 0, = 0. =, =. GEOMETRY The vertices of a quadrilateral are A(, ), B(, ), C(, ), and D(, ). How can ou use slope to determine whether the quadrilateral is a parallelogram? Is it a parallelogram? Justif our answer. Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC MSCC_ALG_PE_00_et.indd /7/ ::8 PM

18 COMMON CORE Parallel and Perpendicular Lines In this etension, ou will identif parallel and perpendicular lines. Preparing for Standards F.IF. F.IF. Perpendicular Lines and Slope Two lines in the same plane that intersect to form right angles are perpendicular lines. Two nonvertical lines are perpendicular if and onl if the product of their slopes is. Vertical lines are perpendicular to horizontal lines. EXAMPLE Identifing Perpendicular Lines Which two lines are perpendicular? How do ou know? Find the slope of each line. Blue Line Red Line Green Line (, ) (0, ) (, ) slope = slope = slope = = ( ) =, or = 0 ( ) = 7 ( ) = 0 ( ) = 7 (, 0) 7 (, ) (, ) The slope of the red line is 7. The slope of the green line is 7. Because 7 7 =, the red and green lines are perpendicular. Which lines are perpendicular? How do ou know? (0, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (0, ) (, ) Are the given lines perpendicular? Eplain our reasoning. 9. =, = 8 0. = 8, = 7. = 0, = 0. GEOMETRY The vertices of a parallelogram are J (, 0), K(, ), L(, ), and M(, ). How can ou use slope to determine whether the parallelogram is a rectangle? Is it a rectangle? Justif our answer. Copright Big Ideas Learning, LLC Etension. Slopes of Parallel and Perpendicular Lines 7

19 . Graphing Linear Equations in Slope-Intercept Form equation = m + b? How can ou describe the graph of the ACTIVITY: Finding Slopes and -Intercepts Work with a partner. Graph the equation. Find the slope of the line. Find the point where the line crosses the -ais. a. = + b. = + c. = d. = + COMMON CORE Graphing Equations In this lesson, ou will fi n dsl o p e sa n d -intercepts of graphs of linear equations. graph linear equations written in slope-intercept form. Learning Standards A.CED. A.REI.0 F.IF. 8 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

20 Math Practice Look for Patterns What patterns do ou notice in the table? What does this tell ou about the graph of the equation? Inductive Reasoning Work with a partner. Graph each equation. Then cop and complete the table. a b Equation Description of Graph Slope of Graph. = + Line. = + Point of Intersection with -ais (0, ) c. = d. = +. = + 7. = 8. = 9. = 0. = +. =. = +. IN YOUR OWN WORDS How can ou describe the graph of the equation = m + b? a. How does the value of m affect the graph of the equation? b. How does the value of b affect the graph of the equation? c. Check our answers to parts (a) and (b) with three equations that are not in the table.. LOGIC Wh do ou think = m + b is called the slope-intercept form of the equation of a line? Use drawings or diagrams to support our answer. Use what ou learned about graphing linear equations in slope-intercept form to complete Eercises on page. Copright Big Ideas Learning, LLC Section. Graphing Linear Equations in Slope-Intercept Form 9

21 . Lesson Lesson Tutorials Ke Vocabular -intercept, p. 0 -intercept, p. 0 slope-intercept form, p. 0 Intercepts The -intercept of a line is the -coordinate of the point where the line crosses the -ais. It occurs when = 0. -intercept = b (0, b) The -intercept of a line is the -coordinate of the point where the line crosses the -ais. It occurs when = 0. O -intercept = a (a, 0) Slope-Intercept Form Words A linear equation written in the form = m + b is in slope-intercept form. The slope of the line is m and the -intercept of the line is b. = m + b Algebra slope EXAMPLE -intercept Identifing Slopes and -Intercepts Find the slope and -intercept of the graph of each linear equation. a. = = + ( ) Write in slope-intercept form. The slope is and the -intercept is. b. = = + Add to each side. The slope is and the -intercept is. Find the slope and -intercept of the graph of the linear equation. Eercises 7 0 Chapter MSCC_ALG_PE_00.indd 0. = 7 Graphing and Writing Linear Equations. = Copright Big Ideas Learning, LLC /9/ :: PM

22 EXAMPLE Graphing a Linear Equation in Slope-Intercept Form Graph = +. Identif the -intercept. Step : Find the slope and -intercept. = + -intercept slope Check Step : The -intercept is. So, plot (0, ). Step : Use the slope to find another point and draw the line. â Ź à Ź = + (0, ) rise slope = = run Plot the point that is unit right and units down from (0, ). Draw a line through the two points. Ź The line crosses the -ais at (, 0). So, the -intercept is. EXAMPLE Real-Life Application The cost (in dollars) of taking a tai miles is =. +. (a) Graph the equation. (b) Interpret the -intercept and slope. a. The slope of the line is. =. Use the slope and -intercept to graph the equation. 7 The -intercept is. So, plot (0, ). Use the slope to plot another point, (, 7). Draw a line through the points. (0, ) 0 0 b. The slope is.. So, the cost per mile is $.0. The -intercept is. So, there is an initial fee of $ to take the tai. Eercises 8 Graph the linear equation. Identif the -intercept. Use a graphing calculator to check our answer.. =. = +. In Eample, the cost (in dollars) of taking a different tai miles is = +.. Interpret the -intercept and slope. Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd Section. Graphing Linear Equations in Slope-Intercept Form /9/ ::8 PM

23 Eercises. Help with Homework. VOCABULARY How can ou find the -intercept of the graph of + =?. CRITICAL THINKING Is the equation = in slope-intercept form? Eplain.. OPEN-ENDED Describe a real-life situation that can be modeled b a linear equation. Write the equation. Interpret the -intercept and slope. )= 9+(- )= +(- 9)= +(- = ) 9+(- Match the equation with its graph. Identif the slope and -intercept. A. B. C.. = +. =. = + Find the slope and -intercept of the graph of the linear equation. 7. = 8. = = 0. = =.. =. + = 8. =. ERROR ANALYSIS Describe and correct the error in finding the slope and -intercept of the graph of the linear equation.. =. + = The slope is and the -intercept is. 7. SKYDIVING A skdiver parachutes to the ground. The height (in feet) of the skdiver after seconds is = Chapter MSCC_ALG_PE_00.indd a. Graph the equation. b. Interpret the -intercept and slope. Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ ::9 PM

24 Graph the linear equation. Identif the -intercept. Use a graphing calculator to check our answer. 8. = + 9. = 7 0. = = =. =. PHONES The cost (in dollars) of making a long distance phone call for minutes is = a. Graph the equation. b. Interpret the slope and -intercept.. APPLES Write a linear equation that models the cost of picking pounds of apples. Graph the equation.. ELEVATOR The basement of a building is 0 feet below ground level. The elevator rises at a rate of feet per second. You enter the elevator in the basement. Write an equation that represents the height (in feet) of the elevator after seconds. Graph the equation. 7. REASONING You work in an electronics store. You earn a fied amount of $ per da, plus a % bonus on the merchandise ou sell. Write an equation that models the amount (in dollars) ou earn for selling dollars of merchandise in one da. Graph the equation. 8. Si friends create a website. The website earns mone b selling banner ads. The site has five banner ads. It costs $0 a month to operate the website. a. A banner ad earns $0.00 per click. Write a linear equation that represents the monthl income (in dollars) for clicks. b. Draw a graph of the equation in part (a). On the graph, label the number of clicks needed for the friends to start making a profit. Solve the equation for. (Section.) 9. = 0. + =. =. 7 + = 8. MULTIPLE CHOICE Which point is a solution of the equation 8 =? (Section.) A (, ) B (, ) C (, ) D (, ) Copright Big Ideas Learning, LLC Section. Graphing Linear Equations in Slope-Intercept Form

25 . Graphing Linear Equations in Standard Form equation a + b = c? How can ou describe the graph of the ACTIVITY: Using a Table to Plot Points Work with a partner. You sold a total of $ worth of tickets to a school concert. You lost track of how man of each tpe of ticket ou sold. $ Adult Number of Adult Tickets + $ Child Number of Child Tickets = $ a. Let represent the number of adult tickets. Let represent the number of child tickets. Write an equation that relates and. b. Cop and complete the table showing the different combinations of tickets ou might have sold. Number of Adult Tickets, Number of Child Tickets, COMMON CORE Graphing Equations In this lesson, ou will graph linear equations written in standard form. Learning Standards A.CED. A.REI.0 F.IF. c. Plot the points from the table. Describe the pattern formed b the points. d. If ou remember how man adult tickets ou sold, can ou determine how man child tickets ou sold? Eplain our reasoning Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

26 ACTIVITY: Rewriting an Equation Math Practice Understand Quantities What do the equation and the graph represent? How can ou use this information to solve the problem? Work with a partner. You sold a total of $ worth of cheese. You forgot how man pounds of each tpe of cheese ou sold. $ Pounds of lb Swiss + $ Pounds of lb Cheddar = $ a. Let represent the number of pounds of Swiss cheese. Let represent the number of pounds of Cheddar cheese. Write an equation that relates and. b. Rewrite the equation in slope-intercept form. Then graph the equation IN YOUR OWN WORDS How can ou describe the graph of the equation a + b = c?. Activities and show two different methods for graphing a + b = c. Describe the two methods. Which method do ou prefer? Eplain.. Write a real-life problem that is similar to those shown in Activities and.. Wh do ou think it might be easier to graph + = 0 using standard form instead of rewriting it in slope-intercept form and then graphing? Use what ou learned about graphing linear equations in standard form to complete Eercises and on page 8. Copright Big Ideas Learning, LLC Section. Graphing Linear Equations in Standard Form

27 . Lesson Lesson Tutorials Ke Vocabular standard form, p. Standard Form of a Linear Equation The standard form of a linear equation is a + b = c Stud Tip where a and b are not both zero. An linear equation can be written in standard form. EXAMPLE Graphing a Linear Equation in Standard Form Graph + =. Step : Write the equation in slope-intercept form. + = Write the equation. = Add to each side. Divide each side b. = Step : Use the slope and -intercept to graph the equation. = + ( ) -intercept slope Check Use the slope to plot another point, (, 0). Ź (0, ). + = 0 Chapter MSCC_ALG_PE_00.indd Draw a line through the points. Graph the linear equation. Use a graphing calculator to check our graph.. + = + = Ź Eercises 0 The -intercept is. So, plot (0, ). Ź à â Ź Graphing and Writing Linear Equations. + =. + = Copright Big Ideas Learning, LLC /9/ ::0 PM

28 EXAMPLE Graphing a Linear Equation in Standard Form Graph + = using intercepts. Step : To find the -intercept, substitute 0 for. To find the -intercept, substitute 0 for. + = + = + (0) = 0 + = = = Step : Graph the equation. Check + = The -intercept is. So, plot (0, ). (, 0) Ź The -intercept is. So, plot (, 0). (0, ) à â Ź Draw a line through the points. Ź EXAMPLE Real-Life Application You have $ to spend on apples and bananas. (a) Graph the equation =, where is the number of pounds of apples and is the number of pounds of bananas. (b) Interpret the intercepts. a. Find the intercepts and graph the equation. -intercept -intercept = =. + 0.(0) =.(0) + 0. = 0 = (0, 0) = 8 = 0 b. The -intercept shows that ou can bu pounds of apples if ou don t bu an bananas. The -intercept shows that ou can bu 0 pounds of bananas if ou don t bu an apples. (, 0) Eercises 8 Graph the linear equation using intercepts. Use a graphing calculator to check our graph.. = 8. + = 7. WHAT IF? In Eample, ou bu pounds of oranges instead of bananas. Oranges cost $.0 per pound. Graph the equation. +. =. Interpret the intercepts. Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd 7 Section. Graphing Linear Equations in Standard Form 7 /9/ :: PM

29 Eercises. Help with Homework. VOCABULARY Is the equation = + in standard form? Eplain.. REASONING Does the graph represent a linear equation? Eplain. 0 0 )= 9+(- )= +(- 9)= +(- = ) 9+(- Define two variables for the verbal model. Write an equation in slope-intercept form that relates the variables. Graph the equation. $.00 pound. miles hour. Pounds of peaches $.0 pound + Hours miles + biked hour Pounds of apples = $ Hours = walked miles Write the linear equation in slope-intercept form.. + = 7. = 7. + = 0 Graph the linear equation. Use a graphing calculator to check our graph = 7 9. = 0. + = Use the graph to find the - and -intercepts..... ERROR ANALYSIS Describe and correct the error in finding the -intercept.. BRACELET A charm bracelet costs $, plus $ for each charm. + = (0) + = = = a. Write an equation in standard form that represents the total cost of the bracelet. b. How much does the bracelet shown cost? 8 Chapter MSCC_ALG_PE_00.indd 8 Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ :: PM

30 Graph the linear equation using intercepts. Use a graphing calculator to check our graph.. = 7. + = 8 8. = 9. SHOPPING The amount of mone ou spend on CDs and DVDs is given b the equation + 8 =. Find the intercepts and graph the equation. Boat: $ 0/da Gear: $ 0/da 0. SCUBA Five friends go scuba diving. The rent a boat for das and scuba gear for das. The total spent is $000. a. Write an equation in standard form that represents the situation. b. Graph the equation and interpret the intercepts.. MODELING You work at a restaurant as a host and a server. You earn $9. for each hour ou work as a host and $7. for each hour ou work as a server. a. Write an equation in standard form that models our earnings. b. Graph the equation. Basic Information Pa to the Order of:... John Doe # of hours worked as... host: # of hours worked as... server: Earnings for this pa... period: $0.. LOGIC Does the graph of ever linear equation have an -intercept? Eplain our reasoning. Include an eample.. For a house call, a veterinarian charges $70, plus $0 an hour. a. Write an equation that represents the total fee charged b the veterinarian for a visit lasting hours. b. Find the -intercept. Will this point appear on the graph of the equation? Eplain our reasoning. c. Graph the equation. Cop and complete the table of values. (Skills Review Handbook) MULTIPLE CHOICE Which value of makes the equation = 9 true? (Section.) A B 0 C D Copright Big Ideas Learning, LLC Section. Graphing Linear Equations in Standard Form 9

31 Stud Help Graphic Organizer You can use a process diagram to show the steps involved in a procedure. Here is an eample of a process diagram for graphing a linear equation. Graphing a linear equation Make a table of values. Eample Graph = 0 Plot the ordered pairs. (, ) (0, ) (, ) Draw a line through the points. (, ) (0, ) (, ) = Make process diagrams with eamples to help ou stud these topics.. finding the slope of a line. graphing a linear equation using a. slope and -intercept b. - and -intercepts After ou complete this chapter, make process diagrams for the following topics.. writing equations in slope-intercept form. writing equations in point-slope form. writing equations of parallel lines. writing equations of perpendicular lines Here is a process diagram with suggestions for what to do if a hena knocks on our door. 70 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

32 .. Quiz Progress Check Graph the linear equation using a table. (Section.). = + 8. =. =. =. Find the slope of the line. (Section.). (, ) (, ). (, ) (, ) 7. (, ) (, ) 8. What is the slope of a line that is parallel to the line in Eercise? What is the slope of a line that is perpendicular to the line in Eercise? (Section.) 9. Are the lines = and = parallel? Are the perpendicular? Justif our answer. (Section.) Find the slope and -intercept of the graph of the linear equation. (Section.) 0. = 8. = + Find the - and -intercepts of the graph of the equation. (Section.). =. + =. BANKING A bank charges $ each time ou use an out-of-network ATM. At the beginning of the month, ou have $00 in our bank account. You withdraw $0 from our bank account each time ou use an out-of-network ATM. Write and graph a linear equation that represents the balance in our account after ou use an out-of-network ATM times. (Section.). STATE FAIR Write a linear equation that models the cost of one person going on rides at the fair. Graph the equation. (Section.). PAINTING You used $90 worth of paint for a school float. (Section.) Admission: $ Rides: $ each a. Graph the equation 8 + = 90, where is the number of gallons of blue paint and is the number of gallons of white paint. b. Interpret the intercepts. Copright Big Ideas Learning, LLC Sections.. Quiz 7

33 . Writing Equations in Slope-Intercept Form How can ou write an equation of a line when ou are given the slope and -intercept of the line? ACTIVITY: Writing Equations of Lines Work with a partner. Find the slope of each line. Find the -intercept of each line. Write an equation for each line. What do the three lines have in common? a. b. 7 c. Chapter MSCC_ALG_PE_00.indd 7 d. 7 Graphing Equations 7 8 COMMON CORE In this lesson, ou will write equations of lines in slope-intercept form. Learning Standards 8.F. A.CED. A.CED. Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ ::8 PM

34 Math Practice Analze Givens What do ou need to know in order to write an equation? Work with a partner. a. ACTIVITY: Describing a Parallelogram Find the area of each parallelogram. Write an equation for each side of each parallelogram. What do ou notice about the slopes of the opposite sides of each parallelogram? b. 7 ACTIVITY: Interpreting the Slope and -Intercept Work with a partner. The graph shows a trip taken b a car where t is the time (in hours) and is the distance (in miles) from Phoeni. a. How far from Phoeni was the car at the beginning of the trip? b. What was the car s speed? c. How long did the trip last? d. How far from Phoeni was the car at the end of the trip? Distance (miles) Car Trip t Time (hours). IN YOUR OWN WORDS How can ou write an equation of a line when ou are given the slope and -intercept of the line? Give an eample that is different from those in Activities,, and. Use what ou learned about writing equations in slope-intercept form to complete Eercises and on page 7. Copright Big Ideas Learning, LLC Section. Writing Equations in Slope-Intercept Form 7

35 . Lesson Lesson Tutorials EXAMPLE Writing Equations in Slope-Intercept Form Write an equation of the line in slope-intercept form. a. Find the slope and -intercept. 7 slope = (0, ) 0 =, or = (, ) Stud Tip Because the line crosses the -ais at (0, ), the -intercept is. After writing an equation, check that the given points are solutions of the equation. slope -intercept So, the equation is = +. b. Find the slope and -intercept. (, ) slope = 0 =, or = (0, ) Because the line crosses the -ais at (0, ), the -intercept is. -intercept slope So, the equation is = + ( ), or =. Write an equation of the line in slope-intercept form. Eercises 0.. (0, ) 7 Chapter MSCC_ALG_PE_00.indd 7 (, ) (0, ) Graphing and Writing Linear Equations (, ) Copright Big Ideas Learning, LLC /9/ :7:00 PM

36 EXAMPLE Writing an Equation Which equation is shown in the graph? A = C =0 Remember B = D = (, ) The line is horizontal, so the change in is 0. change in change in Find the slope and -intercept. The graph of = a is a horizontal line that passes through (0, a). (0, ) 0 slope = = = 0 Because the line crosses the -ais at (0, ), the -intercept is. So, the equation is = 0 + ( ), or =. The correct answer is A. Real-Life Application The graph shows the distance remaining to complete a tunnel. (a) Write an equation that represents the distance (in feet) remaining after months. (b) How much time does it take to complete the tunnel? a. Find the slope and -intercept. change in 000 slope = = = 00 change in Engineers used tunnel boring machines like the ones shown above to dig an etension of the Metro Gold Line in Los Angeles. The new tunnels are.7 miles long and feet wide. Tunnel Digging Distance remaining (feet) EXAMPLE (0, 00) (, 00) Because the line crosses the -ais at (0, 00), the -intercept is Time (months) So, the equation is = b. The tunnel is complete when the distance remaining is 0 feet. So, find the value of when = 0. = Write the equation. 0 = Substitute 0 for. 00 = 00 7= Subtract 00 from each side. Solve for. It takes 7 months to complete the tunnel.. Write an equation of the line that passes through (0, ) and (, ). Eercises Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd 7. WHAT IF? In Eample, the points are (0, 00) and (, 00). How long does it take to complete the tunnel? Section. Writing Equations in Slope-Intercept Form 7 /9/ :7:0 PM

37 Eercises. Help with Homework. PRECISION Eplain how to find the slope of a line given the intercepts of the line.. WRITING Eplain how to write an equation of a line using its graph. )= 9+(- )= +(- 9)= +(- = ) 9+(- Write an equation for each side of the figure... Write an equation of the line in slope-intercept form.. (0, ) (, ) Chapter MSCC_ALG_PE_00.indd 7 Graphing and Writing Linear Equations (, ). BOA A boa constrictor is 8 inches long at birth and grows 8 inches per ear. Write an equation that represents the length (in feet) of a boa constrictor that is ears old. (0, ) (, ). ERROR ANALYSIS Describe and correct the error in writing the equation of the line (0, ) (0, 0) (, ) (0, ) 7. (, ) (, ). (0, ) = + (, 0) (0, ) Copright Big Ideas Learning, LLC /9/ :7:0 PM

38 Write an equation of the line that passes through the points.. (, ), (0, ). (, 0), (0, 0). (0, ), (, ). WALKATHON One of our friends gives ou $0 for a charit walkathon. Another friend gives ou an amount per mile. After miles, ou have raised $.0 total. Write an equation that represents the amount of mone ou have raised after miles. 7. BRAKING TIME During each second of braking, an automobile slows b about 0 miles per hour. a. Plot the points (0, 0) and (, 0). What do the points represent? b. Draw a line through the points. What does the line represent? c. Write an equation of the line. 8. PAPER You have 00 sheets of notebook paper. After week, ou have 7% of the sheets left. You use the same number of sheets each week. Write an equation that represents the number of pages remaining after weeks. 9. The palm tree on the left is 0 ears old. The palm tree on the right is 8 ears old. The trees grow at the same rate. a. Estimate the height (in feet) of each tree. b. Plot the two points (, ), where is the age of each tree and is the height of each tree. c. What is the rate of growth of the trees? d. Write an equation that represents the height of a palm tree in terms of its age. ft Plot the ordered pair in a coordinate plane. (Skills Review Handbook) 0. (, ). (, ). (0, ). (, 7). MULTIPLE CHOICE Which of the following statements is true? (Section.) A The -intercept is. B The -intercept is. C The -intercept is. D The -intercept is. Copright Big Ideas Learning, LLC Section. Writing Equations in Slope-Intercept Form 77

39 . Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the line that has the given slope and passes through the given point. Find the -intercept of the line. Write an equation of the line. a. m = b. m = COMMON CORE Writing Equations In this lesson, ou will write equations of lines using a slope and a point. write equations of lines using two points. Learning Standards A.CED. A.REI.0 F.IF. F.IF. c. m = 7 d. m = 7 78 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

40 Math Practice Construct Arguments How does a graph help ou develop a formula? ACTIVITY: Developing a Formula Work with a partner. a. Draw a nonvertical line that passes through the point (, ). ( b. Plot another point on our line. Label this point, ) as (, ). This point represents an other point on O the line. c. Label the rise and run of the line through the points (, ) and (, ). d. The rise can be written as. The run can be written as. Eplain wh this is true. e. Write an equation for the slope m of the line using the epressions from part (d). f. Multipl each side of the equation b the epression in the denominator. Write our result. What does this result represent? ACTIVITY: Writing an Equation Work with a partner. For months, ou have saved $ a month. You now have $7 in our savings account. Draw a graph that shows the balance in our account after t months. Use our result from Activit to write an equation that represents the balance A after t months. Balance (dollars) Savings Account A t Time (months). Redo Activit using the formula ou found in Activit. Compare the results. What do ou notice?. The formula ou found in Activit, = m( ), is called the point-slope form of the equation of a line. Wh is = m( ) called the point-slope form? Wh do ou think it is important?. IN YOUR OWN WORDS How can ou write an equation of a line when ou are given the slope and a point on the line? Give an eample that is different from those in Activit. Use what ou learned about writing equations using a slope and a point to complete Eercises on page 8. Copright Big Ideas Learning, LLC Section. Writing Equations In Point-Slope Form 79

41 . Lesson Lesson Tutorials Ke Vocabular point-slope form, p. 80 Point-Slope Form A linear equation written in the form = m( ) is in point-slope form. The line passes through the point (, ) and the slope of the line is m. Words slope (, ) Ź = m( ) Algebra (, ) Ź passes through (, ) EXAMPLE O Writing an Equation Using a Slope and a Point Write in point-slope form an equation of the line that passes through the point (, ) with slope. = m( ) Write the point-slope form. Substitute for m, for, and for. Simplif. = [ ( )] = ( + ) So, the equation is = ( + ). Check Check that (, ) is a solution of the equation. = ( + ) Write the equation.? = ( + ) Substitute. 0=0 Eercises Chapter MSCC_ALG_PE_00.indd 80 Simplif. Write in point-slope form an equation of the line that passes through the given point and has the given slope.. (, ); m = 80. (7, 0); m = Graphing and Writing Linear Equations. ( 8, ); m = Copright Big Ideas Learning, LLC /9/ :8:8 PM

42 EXAMPLE Writing an Equation Using Two Points Write in slope-intercept form an equation of the line that passes through the points (, ) and (, ). Stud Tip You can use either of the given points to write the equation of the line. Use m = and (, ). ( ) = ( ) + = + 0 = + 8 Then use the slope m = and the point (, ) to write an equation of the line. = m( ) Write the point-slope form. = ( ) Substitute for m, for, and for. = + Use Distributive Propert. = + 8 EXAMPLE = = = Find the slope: m = Write in slope-intercept form. Real-Life Application You finish parasailing and are being pulled back to the boat. After seconds, ou are feet above the boat. (a) Write and graph an equation that represents our height (in feet) above the boat after seconds. (b) At what height were ou parasailing? a. You are being pulled down at the rate of 0 feet per second. So, the slope is 0. You are feet above the boat after seconds. So, the line passes through (, ). Use the point-slope form. 0 feet per second = 0( ) Substitute for m,, and. = Use Distributive Propert. = 0 + Write in slope-intercept form. So, the equation is = 0 +. â Ź0 à 0 b. You start descending when = 0. The -intercept is. So, ou were parasailing at a height of feet. 0 (, ) 0 0 Ź Ź Eercises 7 7 Write in slope-intercept form an equation of the line that passes through the given points.. (, ), (, ). (, ), (, ). ( 8, ), (, 9) 7. WHAT IF? In Eample, ou are feet above the boat after seconds. Write and graph an equation that represents our height (in feet) above the boat after seconds. Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd 8 Section. Writing Equations In Point-Slope Form 8 /9/ :8:0 PM

43 Eercises. Help with Homework. VOCABULARY From the equation = ( + ), identif the slope and a point on the line.. WRITING Describe how to write an equation of a line using (a) its slope and a point on the line, and (b) two points on the line. )= 9+(- )= +(- 9)= +(- = ) 9+(- Use the point-slope form to write an equation of the line with the given slope that passes through the given point.. m =. m =. m = Write in point-slope form an equation of the line that passes through the given point and has the given slope. 7. (, 8); m =. (, 0); m = 7 0. (, ); m = 9. (7, ); m = 8. (, ); m =. (, ); m = Write in slope-intercept form an equation of the line that passes through the given points.. (, ), (, ). (, ), (8, ). (, ), (, ). (, ), (, 7). ( 9, ), (, ) 7. (, ), (, ) 8. CHEMISTRY At 0 C, the volume of a gas is liters. For each degree the temperature T (in degrees Celsius) increases, the volume V (in liters) of the gas increases b. Write an equation that represents the volume of the gas in terms of the temperature. 8 Chapter MSCC_ALG_PE_00.indd 8 Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ :8: PM

44 C = π 9. CARS After it is purchased, the value of a new car decreases $000 each ear. After ears, the car is worth $8,000. a. Write an equation that represents the value V (in dollars) of the car ears after it is purchased. b. What was the original value of the car? 0. CIRCUMFERENCE Consider the circles shown. a. Plot the points (, π) and (, π). b. Write an equation of the line that passes through the two points. C = π. CRICKETS According to Dolbear s Law, ou can predict the temperature T (in degrees Fahrenheit) b counting the number of chirps made b a snow tree cricket in minute. For each rise in temperature of 0. F, the cricket makes an additional chirp each minute. a. A cricket chirps 0 times in minute when the temperature is 0 F. Write an equation that represents the temperature in terms of the number of chirps in minute. b. You count 00 chirps in minute. What is the temperature? c. The temperature is 9 F. How man chirps would ou epect the cricket to make? Leaning Tower of Pisa (0.7, ). WATERING CAN You water the plants in our classroom at a constant rate. After seconds, our watering can contains 8 ounces of water. Fifteen seconds later, the can contains 8 ounces of water. a. Write an equation that represents the amount (in ounces) of water in the can after seconds. b. How much water was in the can when ou started watering the plants? c. When is the watering can empt? 7.7 m. Problem Solving The Leaning Tower of Pisa in Ital was built between 7 and 0. a. Write an equation for the ellow line. b. The tower is meters tall. How far off center is the top of the tower? Find the percent of the number. (Skills Review Handbook). % of 00. 0% of. % of 7 7. MULTIPLE CHOICE What is the -intercept of the equation + = 0? (Section.) A 0 B C D 0 Copright Big Ideas Learning, LLC Section. Writing Equations In Point-Slope Form 8

45 Etension. Writing Equations of Parallel and Perpendicular LInes Lesson Tutorials You can use the slope-intercept form or the point-slope form to write equations of parallel and perpendicular lines. Remember EXAMPLE Lines that are parallel have the same slope. Lines that are perpendicular have slopes whose product is. Writing an Equation of a Parallel Line Write an equation of the line that passes through (, ) and is parallel to the line = +. Step : Find the slope of the parallel line. The slope of = + is. So, the parallel line that passes through (, ) has the same slope,. Check Step : Use the slope and the slope-intercept form to find the -intercept of the parallel line that passes through (, ). 8 = m + b Write the slope-intercept form. = () + b Substitute for m, for, and for. = + b = b Multipl. Subtract from each side. The parallel line has a slope of and a -intercept of. So, an equation of the parallel line is = + ( ), or =. Write an equation of the line that passes through the given point and is parallel to the given line. Use a graphing calculator to check our answer.. (, ); =. (, ); = +. (, ); = Write an equation of the line that passes through the given point and is parallel to the line shown in the graph.. (, ). (0, 0). (, 8) 7. (, 7) 8 Chapter Graphing and Writing Linear Equations

46 EXAMPLE Writing an Equation of a Perpendicular Line COMMON CORE Writing Equations In this etension, ou will write equations of parallel and perpendicular lines. Learning Standards A.CED. A.REI.0 F.IF. F.IF. Write an equation of the line that passes through (, ) and is perpendicular to the line shown in the graph. Step : Find the slope of the line in the graph. (0, ) (, ) change in slope = change in = = Step : Find the slope of the perpendicular line. The slope of the line in the graph is. Because =, the slope of the perpendicular line is. Step : Use the slope m = and the point-slope form to write an equation of the perpendicular line that passes through (, ). = m( ) Write the point-slope form. = [ ( )] Substitute for m, for, and for. = + = + 7 Simplif. Add to each side. So, an equation of the perpendicular line is = + 7. Write an equation of the line that passes through the given point and is perpendicular to the given line. 8. (, ); = + 9. (0, ); = 0. (, 7); = Write an equation of the line that passes through the given point and is perpendicular to the line shown in the graph.. (, ). (0, ). (, ). (, ). REASONING Rework Eample using the point-slope form. Rework Eample using the slope-intercept form. Which method do ou prefer? Eplain our reasoning. Copright Big Ideas Learning, LLC Etension. Writing Equations of Parallel and Perpendicular Lines 8

47 .7 Solving Real-Life Problems two variables to model and solve a real-life problem? How can ou use a linear equation in EXAMPLE: Writing a Stor Write a stor that uses the graph at the right. In our stor, interpret the slope of the line, the -intercept, and the -intercept. Make a table that shows data from the graph. Label the aes of the graph with units. Draw pictures for our stor. There are man possible stories. Here is one about a reef tank Tom works at an aquarium shop on Saturdas. One Saturda, when Tom gets to work, he is asked to clean a 7-gallon reef tank. His first job is to drain the tank. He puts a hose into the tank and starts a siphon. Tom wonders if the tank will finish draining before he leaves work. He measures the amount of water that is draining out and finds that. gallons drain out in 0 minutes. So, he figures that the rate is gallons per hour. To see when the tank will be empt, Tom makes a table and draws a graph. COMMON CORE Writing Equations In this lesson, ou will solve real-life problems involving linear equations. Appling Standards 8.F. A.CED. F.IF. -intercept: number of hours to empt the tank intercept: amount of water in full tank From the table and also from the graph, Tom sees that the tank will be empt after 7 hours. This will give him hour to wash the tank before going home. Water (gallons) Time (hours) 8 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

48 ACTIVITY: Writing a Stor Math Practice Label Aes What information is needed to label the aes? How do ou know where to place the labels? Work with a partner. Write a stor that uses the graph of a line. In our stor, interpret the slope of the line, the -intercept, and the -intercept. Make a table that shows data from the graph. Label the aes of the graph with units. Draw pictures for our stor. ACTIVITY: Drawing Graphs Work with a partner. Describe a real-life problem that has the given rate and intercepts. Draw a line that represents the problem. a. Rate: 0 feet per second -intercept: 0 feet -intercept: seconds b. Rate: dollars per month -intercept: $00 -intercept: 8 months IN YOUR OWN WORDS How can ou use a linear equation in two variables to model and solve a real-life problem? List three different rates that can be represented b slopes in real-life problems. Use what ou learned about solving real-life problems to complete Eercises and on page 90. Copright Big Ideas Learning, LLC Section.7 Solving Real-Life Problems 87

49 .7 Lesson Lesson Tutorials EXAMPLE Real-Life Application The percent (in decimal form) of batter power remaining hours after ou turn on a laptop computer is = (a) Graph the equation. (b) Interpret the - and -intercepts. (c) After how man hours is the batter power at 7%? a. Use the slope and the -intercept to graph the equation. = 0. + slope -intercept The -intercept is. So, plot (0, ). (0, ) (, 0.8) Use the slope to plot another point, (, 0.8). Draw a line through the points b. To find the -intercept, substitute 0 for in the equation. = 0. + Write the equation. 0 = 0. + Substitute 0 for. = Solve for. The -intercept is. So, the batter lasts hours. The -intercept is. So, the batter power is at 00% when ou turn on the laptop. c. Find the value of when = 0.7. = 0. + Write the equation. 0.7 = 0. + Substitute 0.7 for.. = Solve for. 7% Remaining The batter power is at 7% after. hours. Eercise. The amount (in gallons) of gasoline remaining in a gas tank after driving hours is = +. (a) Graph the equation. (b) Interpret the - and -intercepts. (c) After how man hours are there gallons left? 88 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

50 EXAMPLE Real-Life Application F 90 The graph relates temperatures (in degrees Fahrenheit) to temperatures (in degrees Celsius). (a) Find the slope and -intercept. (b) Write an equation of the line. (c) What is the mean temperature of Earth in degrees Fahrenheit? change in change in 0 7 (0, ) 0 9 (0, 8) 0 C a. slope = = = The line crosses the -ais at (0, ). So, the -intercept is. 9 The slope is and the -intercept is. b. Use the slope and -intercept to write an equation. slope -intercept 9 The equation is = +. Mean Temperature: íc c. In degrees Celsius, the mean temperature of Earth is. To find the mean temperature in degrees Fahrenheit, find the value of when =. 9 Write the equation. = () + 9 Substitute for. = 9 Simplif. = + The mean temperature of Earth is 9 F. Eercise 7. The graph shows the height (in feet) of a flag seconds after ou start raising it up a flagpole. a. Find and interpret the slope. b. Write an equation of the line. c. What is the height of the flag after 9 seconds? Copright Big Ideas Learning, LLC MSCC_ALG_PE_007.indd 89 Section.7 9 (, ) (0, ) Solving Real-Life Problems 89 /9/ :9:0 PM

51 Eercises.7 Help with Homework. REASONING Eplain how to find the slope, -intercept, and -intercept of the line shown.. OPEN-ENDED Describe a real-life situation that uses a negative slope.. REASONING In a real-life situation, what does the slope of a line represent? )= 9+(- )= +(- 9)= +(- = ) 9+(- Describe a real-life problem that has the given rate and intercepts. Draw a line that represents the problem.. Rate:. gallons per hour. Rate: F per hour -intercept: gallons -intercept: F -intercept: 0 hours -intercept: 7 hours. DOWNLOAD You are downloading a song. The percent (in decimal form) of megabtes remaining to download after seconds is = a. Graph the equation. b. Interpret the - and -intercepts. c. After how man seconds is the download 0% complete? 7. HIKING The graph relates temperature (in degrees Fahrenheit) to altitude (in thousands of feet). a. Find the slope and -intercept. b. Write an equation of the line. c. What is the temperature at sea level? Altitude Change Temperature ( F) 70 0 (0, 9) 0 0 (7,.8) Altitude (thousands of feet) 90 Chapter MSCC_ALG_PE_007.indd 90 Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC /9/ :9:0 PM

52 Distance from St. Louis (miles) 8. REASONING Your famil is driving from Cincinnati to St. Louis. The graph relates our distance from St. Louis (in miles) and travel time (in hours). Driving Distance Springfield 0 0 St. Louis Time (hours) 7 Indianapolis 70 a. Interpret the - and -intercepts. b. What is the slope? What does the slope represent in this situation? c. Write an equation of the line. How would the graph and the equation change if ou were able to travel in a straight line? Daton 7 Cincinnati 9. PROJECT Use a map or the Internet to find the latitude and longitude of our school to the nearest whole number. Then find the latitudes and longitudes of: Antananarivo, Madagascar; Denver, Colorado; Brasilia, Brazil; London, England; and Beijing, China. a. Plot a point for each of the cities in the same coordinate plane. Let the positive -ais represent north and the positive -ais represent east. b. Write an equation of the line that passes through Denver and Beijing. c. In part (b), what geographic location does the -intercept represent? 0. A band is performing at an auditorium for a fee of $00. In addition to this fee, the band receives 0% of each $0 ticket sold. The maimum capacit of the auditorium is 800 people. a. Write an equation that represents the band s revenue R when tickets are sold. b. The band needs $000 for new equipment. How man tickets must be sold for the band to earn enough mone to bu the new equipment? Solve the equation. Check our solution. (Section.). h 7h + =. (k 0) =. 9 +.(q ) = 0. MULTIPLE CHOICE Which equation is the slope-intercept form of 8 =? (Section.) A = + 7 B = 7 C = 7 D = + 7 Section.7 Solving Real-Life Problems 9

53 ..7 Quiz Write an equation of the line in slope-intercept form. (Section.). (, ) (0, ). (0, 0) (, ). Progress Check (0, ) (, ) Write in point-slope form an equation of the line that passes through the given point and has the given slope. (Section.). (, ); m =. (, ); m =. (, ); m = 7. (8, ); m = 8 Write in slope-intercept form an equation of the line that passes through the given points. (Section.) 8. ( 0, ) (, ) 9. (, 0), (0, ) 0. Write an equation of the line that passes through (, ) and is (a) parallel to and (b) perpendicular to the line = +. (Section.). CONSTRUCTION A construction crew is etending a highwa sound barrier that is miles long. The crew builds mile per week. Write an equation for the length (in miles) of the barrier after weeks. (Section.). FISH POND You are draining a fish pond. The amount (in liters) of water remaining after hours is = (a) Graph the equation. (b) Interpret the - and -intercepts. (Section.7). WATER A recreation department bought bottled water to sell at a fair. The graph shows the number of bottles remaining after each hour. (Section.7) a. Find the slope and -intercept. b. Write an equation of the line. c. The fair started at 0 a.m. When did the recreation department run out of bottled water? Bottles remaining Bottled Water 0 (0, 80) (8, 0) Time (hours) 9 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

54 Chapter Review Review Ke Vocabular linear equation p. solution of a linear equation, p. slope, p. 0 rise, p. 0 run, p. 0 perpendicular lines, p. 7 Review Eamples and Eercises -intercept, p. 0 -intercept, p. 0 slope-intercept form, p. 0 standard form, p. point-slope form, p. 80 Vocabular Help. Graphing Linear Equations (pp. 7) Graph =. Step : Make a table of values. = (, ) = ( ) 7 (, 7) = ( ) (, ) 0 = (0) (0, ) = () (, ) Step : Plot the ordered pairs. Step : Draw a line through the points. (, ) (, ) (, ) (0, ) (, ) (0, ) (, 7) 7 (, 7) 7 = Graph the linear equation.. =. =. = 9. =. = +. = Copright Big Ideas Learning, LLC Chapter Review 9

55 . Slope of a Line (pp. 8 7) Find the slope of each line in the graph. Red Line: slope = ( ) = = 8 0 The slope of the red line is undefined. Blue Line: slope = = ( ) = 7, or 7 Green Line: slope = = 0 = 0, or 0 (, ) (, ) (0, ) (, ) (, ) (, ) The points in the table lie on a line. How can ou find the slope of the line from the table? What is the slope? Are the lines = and = parallel? Are the perpendicular? Eplain.. Graphing Linear Equations in Slope-Intercept Form (pp. 8 ) Graph = 0.. Identif the -intercept. Step : Find the slope and -intercept. slope = 0. + ( ) Step : The -intercept is. So, plot (0, ). -intercept Step : Use the slope to find another point and draw the line. slope = rise run = Plot the point that is units right and unit up from (0, ). Draw a line through the two points. The line crosses the -ais at (, 0). So, the -intercept is. = 0. (, 0) (0, ) Graph the linear equation. Identif the -intercept. Use a graphing calculator to check our answer. 0. =. = + 8. = 8 9 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

56 . Graphing Linear Equations in Standard Form (pp. 9) Graph 8 + =. Step : Write the equation in slope-intercept form. 8 + = Write the equation. = 8 + Subtract 8 from each side. = + Divide each side b. Step : Use the slope and -intercept to plot two points. slope = + -intercept The -intercept is. So, plot (0, ). (0, ) (, ) Use the slope to plot another point, (, ). Step : Draw a line through the points. 8 (0, ) (, ) Graph the linear equation.. + =. + = 8. + = = 7. A dog kennel charges $0 per night to board our dog and $ for each hour of pla time. The amount of mone ou spend is given b 0 + = 80, where is the number of nights and is the number of hours of pla time. Graph the equation and interpret the intercepts. Copright Big Ideas Learning, LLC Chapter Review 9

57 . Writing Equations in Slope-Intercept Form (pp. 7 77) Write an equation of the line in slope-intercept form. a. Find the slope and -intercept. (, ) slope = = 0 =, or (0, ) Because the line crosses the -ais at (0, ), the -intercept is. slope -intercept So, the equation is = +, or = +. b. Find the slope and -intercept. (0, ) (, ) slope = ( ) = 0 =, or Because the line crosses the -ais at (0, ), the -intercept is. slope -intercept So, the equation is = + ( ), or =. Write an equation of the line in slope-intercept form (, ) (0, ) (0, ) (, ) 0. Write an equation of the line that passes through (0, 8) and (, 8).. Write an equation of the line that passes through (0, ) and (, ). 9 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

58 . Writing Equations in Point-Slope Form (pp. 78 8) Write in slope-intercept form an equation of the line that passes through the points (, ) and (, ). Find the slope. m = = =, or Then use the slope and one of the given points to write an equation of the line. Use m = and (, ). = m( ) Write the point-slope form. = ( ) Substitute for m, for, and for. = 8 = 7 Use Distributive Propert. Write in slope-intercept form. (, ) (, ) So, the equation is = 7.. Write in point-slope form an equation of the line that passes through the point (, ) with slope.. Write in slope-intercept form an equation of the line that passes through the points (, ) and (, )..7 Solving Real-Life Problems (pp. 8 9) The amount (in dollars) of mone ou have left after plaing games at a carnival is = How much mone do ou have after plaing eight games? = Write the equation. = 0.7(8) + 0 Substitute 8 for. = Simplif. You have $ left after plaing 8 games.. HAY The amount (in bales) of ha remaining after feeding cows for das is = (a) Graph the equation. (b) Interpret the - and -intercepts. (c) How man bales are left after 0 das? Copright Big Ideas Learning, LLC Chapter Review 97

59 Chapter Test Test Practice Find the slope and -intercept of the graph of the linear equation.. =. = 0 +. =. = = = 7 Graph the linear equation. 7. = + 8. = 9. + = 0. Which lines are parallel? Which lines. The points in the table lie on a line. How are perpendicular? Eplain. can ou find the slope of the line from the table? What is the slope? (, ) (, ) (, ) (, ) (, ) (, 0.) (, ) (, ) 0 Write an equation of the line in slope-intercept form.. (, ) (0, 0). (0, ) (, ) Write in slope-intercept form an equation of the line that passes through the given points.. (, ), (, ). (, ), (, ). (, ), (, ) 7. BRAILLE Because of its size and detail, Braille takes longer to read than tet. A person reading Braille reads at % the rate of a person reading tet. a. Write and graph an equation that represents the average rate of a Braille reader in terms of the average rate of a tet reader. b. Interpret the solution (80, ). c. What happens to as increases? Eplain. 98 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC

60 Standards Assessment. Which equation matches the line shown in the graph? (A.REI.0) Test-Taking Strateg Estimate the Answer A. = B. = + C. = D. = + O Using estimation ou can see that there are about 00 hairs. So, it s got to be C.. A line has a slope of and passes through the point (a, b). Which point must also lie on this line? (F.IF.) F. (a, b + ) H. (a +, b + ) G. (a, 8b) I. (a, b). A car s value depreciates at a rate of $,00 per ear. Three ears after it was purchased, the car s value was $,000. Which equation can be used to find v, its value in dollars, n ears after it was purchased? (A.CED.) A. v = 8,00,00n C. v = 8,00,00n B. v =,000,00n D. v = 8,00 n. The equation = is written in standard form. Which point lies on the graph of this equation? (A.REI.0) F. (, ) H. (, ) G. (, ) I. (, ) Copright Big Ideas Learning, LLC Standards Assessment 99

61 . The line shown in the graph below has a slope of. What is the equation of the line? (A.REI.0) (, ) O A. = 0 C. = + B. = + 0 D. =. A cell phone plan costs $0 per month plus $0.0 for each minute used. Last month, ou spent $8.0 using this plan. This can be modeled b the equation below, where m represents the number of minutes used. 0.m + 0 = 8. How man minutes did ou use last month? (A.REI.) F. 8. min H. 8 min G. 8 min I. 8 min 7. What is the slope of the line that passes through the points (, ) and (8, )? (F.IF.) 8. It costs $0 to rent a car for one da. In addition, the rental agenc charges ou for each mile driven, as shown in the graph. (F.IF.) Part A Determine the slope of the line joining the points on the graph. The Cost of Renting a Car 00 (00, 90) 80 (00, 70) 0 (00, 0) (00, 80) (00, 0) 0 (0, 0) 0 0 Part B Eplain what the slope represents Rental cost (dollars) Miles driven 00 Chapter Graphing and Writing Linear Equations Copright Big Ideas Learning, LLC