Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five friends. Two of ou bu a ticket and a bag of popcorn. The rest of our friends bu just one ticket. The epression + + represents the situation. Evaluate the epression when ( ) tickets cost $7. and a bag of popcorn costs $.. Copright Big Ideas Learning, LLC
Name Date Chapter Fair Game Review (continued) Use the graph to answer the question. 0. Write the ordered pair that corresponds to Point D. F A. Write the ordered pair that corresponds to Point H. D G. Which point is located at (, )? E C B H. Which point is located at ( 0, )?. Which point(s) are located in Quadrant IV?. Which point(s) are located in Quadrant III? Plot the point.. (, ) 7. ( 0, ) 8. (, ) 9. (, 0) 0. (, ) Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations For use with Activit. Essential Question How can ou recognize a linear equation? How can ou draw its graph? 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. = + Solution Points c. 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. GENERALIZE Do ou think it is true that an point on the line is a solution point of the equation = +? Eplain. Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations (continued) f. Choose five additional -values for the table. (Choose positive and negative -values.) Plot the five corresponding solution points on the previous page. Does each point lie on the line? Solution Points = + g. GENERALIZE Do ou think it is true that an solution point of the equation = + is a point on the line? Eplain. h. THE MEANING F A WRD Wh is = a + b called a linear equation? HISTRY: Analtic Geometr Rene Descartes was a French philosopher, scientist, and mathematician. Up until the time of Descartes, algebra and geometr were separate fields of mathematics. Descartes s invention of the coordinate plane was of huge importance to mathematics. For the first time, people could see solutions of equations. No longer did people have to work with algebra from a purel smbolic point of view. Rene Descartes (9 0) Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations (continued) Descartes s combination of geometr and algebra is called analtic (or algebraic) geometr. ne of the main discoveries in analtic geometr is that all of the important tpes of graphs (lines, parabolas, circles, ellipses, and so on) can be represented b simple algebraic equations. Within a few dozen ears, other mathematicians were able to discover all of calculus, a field of mathematics that is of great value in business, science, and engineering. In this book, ou will stud lines. In Algebra and Algebra, ou will stud man other tpes of equations. Line: = a + b Parabola: = a + b Circle: + = r What Is Your Answer?. IN YUR WN WRDS 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.. Are ou a visual learner? Most people can learn mathematics more easil when the see pictures of the mathematics. Wh do ou think Descartes s invention was important to mathematics? Copright Big Ideas Learning, LLC 7
Name Date. Practice For use after Lesson. Graph the linear equation.. =. = Solve for. Then graph the equation.. + =. =. The equation = + represents the cost (in dollars) of renting a movie after das of late charges. a. Graph the equation. b. Use the graph to determine how much it costs after das of late charges. 8 Copright Big Ideas Learning, LLC
Name Date. Slope of a Line For use with Activit. Essential Question How can the slope of a line be used to describe the line? 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). slope = change in 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 gra points. Do ou get the same slope using each method? a. b. c. d. Copright Big Ideas Learning, LLC 9
Name Date. Slope of a Line (continued) ACTIVITY: Drawing Lines with Given Slopes Work with a partner. Draw a line through the black point using the given slope. Draw a line through the gra point using the given slope. What do ou notice about the two lines? a. Slope = b. Slope = c. Slope = d. Slope = What Is Your Answer?. IN YUR WN WRDS How can the slope of a line be used to describe the line? 0 Copright Big Ideas Learning, LLC
Name Date. Slope of a Line (continued) a. Draw three lines that have b. Draw three lines that have positive slopes. negative slopes.. Line A has a slope of. Line B has. Line C has a slope of. Line D a slope of. Compare the slopes of has a slope of. Compare the slopes the lines. Illustrate our comparison. of the lines. Illustrate our comparison. Copright Big Ideas Learning, LLC
Name Date Practice. For use after Lesson. Find the slope of the line... 9 (, ) (, 9) (, ) 9 (, ) The points in the table lie on a line. Find the slope of the line. Then draw its graph.. 0 0. 0 7. The back of a moving truck is feet off of the ground. A ramp etending from the back of the truck meets the ground 0 feet from the truck. What is the slope of the ramp?. Which set of stairs is more difficult to walk up? Eplain. 0 in. in. in. 8 in. Staircase Staircase Copright Big Ideas Learning, LLC
Name Date.b Practice For use after Lesson.b Tell whether the two right triangles are similar. Eplain our reasoning.. Y. G B A C 8 S R T X Z F 8 H. The graph shows similar right triangles drawn using pairs of points on a line. 0 9 8 7 D(8, 8) B(, ) A(, ) C(, ) E(0, 0) F(0, 8) 7 8 9 0 a. For each triangle, find the ratio of the length of the vertical leg to the length of the horizontal leg. b. Relate the ratios in part (a) to the slope of the line. Copright Big Ideas Learning, LLC A
Name Date.b Practice (continued). Consider the line shown in the graph. a. Draw two triangles that show the rise and the run of the line using points A and B and points R and S. b. Use the triangles to find the slope of the line. 0 9 8 7 A(0, ) R(, 8) S(0, ) B(, ) 7 8 9 0 c. Repeat parts (a) and (b) using different pairs of points. A(0, ) 0 9 8 7 R(, 8) S(0, ) B(, ) 7 8 9 0 B Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations in Slope-Intercept Form For use with Activit. Essential Question How can ou describe the graph of the equation = m + b? ACTIVITY: Finding Slopes and -Intercepts Work with a partner. a. Graph the equation. Find the slope of the line. Find the point where the line crosses the -ais. = + b. = + c. = d. = + Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations in Slope-Intercept Form (continued) Inductive Reasoning Work with a partner. Graph each equation. Then complete the table. Equation Description of Graph Slope of Graph Point of Intersection with -ais a. = + b. = + c. = d. = +. = + 7. = 8. 9. = = 0. = +. =. = + Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations in Slope-Intercept Form (continued) What Is Your Answer?. IN YUR WN WRDS 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.. Wh is = m + b called the slope-intercept form of the equation of a line? Copright Big Ideas Learning, LLC
Name Date. Practice For use after Lesson. Find the slope and -intercept of the graph of the linear equation.. = + 9. =. + = 8 Graph the linear equation. Identif the -intercept.. = +. 0 =. The equation = 90 + 0 represents the time (in minutes) left after games of a tournament. a. Graph the equation. b. Interpret the -intercept and slope. Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations in Standard Form For use with Activit. Essential Question How can ou describe the graph of the equation a + b = c? 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 $ Number of + = Adult Tickets Child Child Tickets $ a. Let represent the number of adult tickets. Let represent the number of child tickets. Write an equation that relates and. b. Complete the table showing the different combinations of tickets ou might have sold. Number of Adult Tickets, Number of Child Tickets, c. Plot the points from the table. Describe the pattern formed b the points. 9 8 7 d. If ou remember how man adult tickets ou sold, can ou determine how man child tickets ou sold? Eplain our reasoning. 0 0 7 8 9 Copright Big Ideas Learning, LLC 7
Name Date. Graphing Linear Equations in Standard Form (continued) ACTIVITY: Rewriting an Equation 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 Swiss lb + $ Pounds of Cheddar lb = $ 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. Write the equation in slope-intercept form. Then graph the equation. 9 8 7 0 8 0 7 8 9 Copright Big Ideas Learning, LLC
Name Date. Graphing Linear Equations in Standard Form (continued) What Is Your Answer?. IN YUR WN WRDS 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. Copright Big Ideas Learning, LLC 9
Name Date. Practice For use after Lesson. Write the linear equation in slope-intercept form.. = 7. + =. = 0 7 Graph the linear equation using intercepts.. =. + 9 = 7. You go shopping and bu shirts for $ and jeans for $8. The total spent is $8. a. Write an equation in standard form that models how much mone ou spent. b. Graph the equation and interpret the intercepts. 0 Copright Big Ideas Learning, LLC
Name Date. Sstems of Linear Equations For use with Activit. Essential Question How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast in our home. You spend $00 fiing up a bedroom to rent. Your cost for food and utilities is $0 per night. Your famil charges $0 per night to rent the bedroom. a. Write an equation that represents our costs. Cost, C (in dollars) = $0 per night Number of nights, + $00 b. Write an equation that represents our revenue (income). Revenue, R (in dollars) = $0 per night Number of nights, c. A set of two (or more) linear equations is called a sstem of linear equations. Write the sstem of linear equations for this problem. Copright Big Ideas Learning, LLC
Name Date. Sstems of Linear Equations (continued) ACTIVITY: Using a Table to Solve a Sstem Use the cost and revenue equations from Activit to find how man nights ou need to rent the bedroom before ou recover the cost of fiing up the bedroom. This is the break-even point for our business. a. Complete the table. 0 7 8 9 0 C R b. How man nights do ou need to rent the bedroom before ou break even? ACTIVITY: Using a Graph to Solve a Sstem a. Graph the cost equation from Activit. 00 b. In the same coordinate plane, graph the revenue equation from Activit. 00 00 00 c. Find the point of intersection of the two graphs. The -value of this point is the number of nights ou need to rent the bedroom to break even. 00 00 0 0 7 8 9 0 Copright Big Ideas Learning, LLC
Name Date. Sstems of Linear Equations (continued) ACTIVITY: Using an Equation to Solve a Sstem a. Write the cost equation from Activit. C = b. Write the revenue equation from Activit. R = c. The break-even point occurs when C = R. Set the epression for C equal to the epression for R. You should obtain an equation with on both sides. Solve this equation for. The solution is our break-even point. d. Did ou obtain the same break-even point in Activities,, and? If not, check our work. The break-even point should be the same in all three activities. What Is Your Answer?. IN YUR WN WRDS How can ou solve a sstem of linear equations?. When solving a sstem of linear equations, eplain wh it is a good idea to use two different was to find the solution. Copright Big Ideas Learning, LLC
Name Date. Practice For use after Lesson. Solve the sstem of linear equations using a table.. = +. = 0. = = + 0 Solve the sstem of linear equations using a graph.. = +. = = = 8 Solve the sstem of linear equations algebraicall.. = +. = 7 = + = 7. There are students in our class. There are more girls than bos. Use a sstem of linear equations to find how man bos are in our class. How man girls are in our class? Copright Big Ideas Learning, LLC
Name Date. Special Sstems of Linear Equations For use with Activit. Essential Question Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your cousin is ears older than ou. Your ages can be represented b two linear equations. = t Your age = t + Your cousin s age a. Graph both equations in the same coordinate plane. b. What is the vertical distance between the two graphs? What does this distance represent? 0 8 0 0 8 0 t c. Do the two graphs intersect? If not, what does this mean in terms of our age and our cousin s age? ACTIVITY: Using a Table to Solve a Sstem Work with a partner. You invest $00 for equipment to make dog backpacks. Each backpack costs ou $ for materials. You sell each backpack for $. a. Complete the table for our cost C and our revenue R. 0 7 8 9 0 C R Copright Big Ideas Learning, LLC
Name Date. Special Sstems of Linear Equations (continued) b. When will our compan break even? What is wrong? ACTIVITY: Using a Graph to Solve a Puzzle Work with a partner. Let and be two numbers. Here are two clues about the values of and. Clue : Clue : Words The value of is more than twice the value of. The difference of and is. Equation = + = a. Graph both equations in the same coordinate plane. b. Do the two lines intersect? Eplain. c. What is the solution of the puzzle? d. Use the equation = + to complete the table. 0 7 8 9 0 Copright Big Ideas Learning, LLC
Name Date. Special Sstems of Linear Equations (continued) e. Does each solution in the table satisf both clues? f. What can ou conclude? How man solutions does the puzzle have? How can ou describe them? What Is Your Answer?. IN YUR WN WRDS Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? Give eamples to support our answers. Copright Big Ideas Learning, LLC 7
Name Date. Practice For use after Lesson. Solve the sstem of linear equations.. =. + = 0 = + 7 =. = +. = 8 + = 9 =. You start reading a book for our literature class two das before our friend. You both read 0 pages per night. A sstem of linear equations that represents this situation is = 0 + 0 and = 0. Will our friend finish the book before ou? Eplain.. You and a friend bu music from different online stores. The sstem of linear equations represents the total amount spent on songs and albums. Both of ou bu the same number of albums. Compare the number of songs each of ou bu. 0.9 + 0 = 9..9 + 0 = 79 8 Copright Big Ideas Learning, LLC
Name Date.7 Solving Equations b Graphing For use with Activit.7 Essential Question How can ou use a sstem of linear equations to solve an equation that has variables on both sides? You learned how to use algebra to solve equations with variables on both sides. Another wa is b using a sstem of linear equations. ACTIVITY: Solving a Sstem of Linear Equations Work with a partner. Find the solution of = +. a. Use the left side of the equation to write one linear equation. Then, use the right side to write another linear equation. = + = = + b. Sketch the graphs of the two linear equations. Find the -value of the point of intersection. The -value is the solution of = +. Check the solution. c. Eplain wh this graphical method works. ACTIVITY: Using a Graphing Calculator Use a graphing calculator to graph the two linear equations. = = + The steps used to enter the equations depend on the calculator model that ou have. = = + Copright Big Ideas Learning, LLC 9
Name Date.7 Solving Equations b Graphing (continued) ACTIVITY: Using a Sstem of Linear Equations Work with a partner. Solve the equation using two methods. a. Method : Use an algebraic method. Method : Use a graphical method. Is the solution the same using both methods? + = + b. + = + c. = d. 7 + = 7 0 Copright Big Ideas Learning, LLC
Name Date.7 Solving Equations b Graphing (continued) What Is Your Answer?. IN YUR WN WRDS How can ou use a sstem of linear equations to solve an equation that has variables on both sides? Give an eample that is different from those in Activities and.. Describe three was in which Rene Descartes s invention of the coordinate plane allows ou to solve algebraic problems graphicall. Copright Big Ideas Learning, LLC
Name Date.7 Practice For use after Lesson.7 Solve the equation algebraicall and graphicall.. + =. + = 7 + 8 7 7 7 8 7 8 Use a graph to solve the equation. Check our solution.. + =. + =. n the first da of our garage sale, ou earned + 9 dollars. The net da ou earned dollars. Is it possible that ou earned the same amount each da? Eplain. Copright Big Ideas Learning, LLC