Linear Equations and Systems

Transcription

1 UNIT Linear Equations and Sstems You can use a linear equation to describe a car s speed or the distance it travels.

2 Anthing involving a constant rate can be described with a linear equation. You can find constant rates when driving a car, calculating a cost, or getting paid an hourl wage. Big Ideas A famil of functions has the same general form. The values of the parameters in the equation create transformed versions of other famil members. Unit Topics Foundations for Unit Graphs of Lines Forms of Linear Equations Writing Equations of Lines Applications: Linear Equations Sstems of Linear Equations Applications: Linear Sstems LINEAR EQUATIONS AND SYSTEMS 7

3 Foundations for Unit Before ou stud linear equations and sstems, ou should know how to do the following: Plot a point in a coordinate plane. Find the coordinates of a point. Identif the quadrants of a coordinate plane. Ordered Pairs in a Coordinate Plane Definition coordinate plane a plane in which the coordinates of an point are the point's distances from two intersecting perpendicular lines called aes In a coordinate plane, the perpendicular aes are number lines. The horizontal ais is the -ais and the vertical ais is the -ais. The intersection of the aes is the origin. An ordered pair shows the coordinates of a point in the coordinate plane. In an ordered pair, the -coordinate is the first number and the -coordinate is the second number. For eample, in the ordered pair (7, ), 7 is the -coordinate and is the -coordinate. 7 is the -coordinate is the -coordinate TIP Unless otherwise noted, coordinate plane means rectangular coordinate plane, and the aes are horizontal and vertical. TIP In an ordered pair, the -coordinate comes before the -coordinate; in the alphabet, X comes before Y. (7, ) A point is the graph of an ordered pair, and an ordered pair contains the coordinates of a point. Eample Locate each point in the same coordinate plane: A (, 6), B (, ), C (0, ) Solution To locate -coordinates, move from the origin to the right for positive values and to the left for negative values. To locate -coordinates, move up for positive values and down for negative values. A (, 6): Start at the origin. Move units to the right, and then move 6 units up. B (, ): Start at the origin. Move units to the left, and then move unit down. C (0, ): Start at the origin. Move 0 units to the right or left (sta at the origin), and then move 5 units down B(, ) C(0, ) 6 7 A(, 6) 8 UNIT LINEAR EQUATIONS AND SYSTEMS

4 Unit Foundations Problem Set A Plot each ordered pair.. A (, ) 5. E (, ) 9. T (.9,.). B (0, 5) 6. F (, ) 0. U (, 6 ). C (, 5) 7. R (., 0). V (0,.7). D (, 0) 8. S (, 6 ). W (,.5 ) Finding the Coordinates of a Point Find the coordinates of a point b starting at the origin and counting left or right and up or down. Eample Identif the coordinates of each point shown in the coordinate plane. Solution Point E: From the origin, point E is units right and 5 units down. The coordinates are (, ). Point F: From the origin, point F is units right and 0 units up or down. The coordinates are (, 0). Point G: From the origin, point G is units left and units down. The coordinates are (, ) F G 6 7 E Problem Set B Identif the coordinates of each point.. A 9. R. B 0. S 5. C. T 6. D. U 7. E. V 8. F. W 7 W 6 5 R A F S B E D C T U 6 V 7 FOUNDATIONS FOR UNIT 9

5 Unit Foundations Identifing Quadrants The - and -aes divide the coordinate plane into four regions called quadrants. The are numbered I, II, III, and IV, starting at the top right and moving counterclockwise. -ais 5 Quadrant II the origin Quadrant III Quadrant I 5 Quadrant IV -ais TIP Knowing the signs of the coordinates in each quadrant can help ou locate points. QI: (+, +) QII: (, +) QIII: (, ) QIV: (+, ) Ecept for the origin, ever point in the coordinate plane is located either in one of the quadrants or on one of the aes. The origin is located on both aes. Problem Set C For each point, name the ais it is on or the quadrant it lies in. 5. A (, ) 9. E (, ). T (.9,.) 6. B (0, 5) 0. F (, ). U (, 6 ) 7. C (, 5). R (., 0) 5. V (0,.7) 8. D (, 0). S (, 6 ) 6. W (,.5 ) Answer each question. 7. In which quadrant(s) is the -coordinate negative? 8. In which quadrant(s) is the -coordinate positive? 9. In which quadrant(s) is the -coordinate negative? 0. In which quadrant(s) is the -coordinate positive? 50 UNIT LINEAR EQUATIONS AND SYSTEMS

6 Graphs of Lines You can graph a line in the coordinate plane. Graphing a Linear Equation A linear equation is an equation whose graph in a coordinate plane is a line. You can graph a linear equation b plotting two ordered pairs that make the equation true, and then drawing a line through those points. But, to guard against errors, it is best to plot three or more ordered pairs. Eample Graph the linear equation = + 6. Solution Use a table to find three ordered pairs that are solutions of the equation. Pick an three -values and calculate the corresponding -values How to calculate the -value: = = 6 = + 6 = = + 6 = (0, 6) (, ) = + 6 (5, ) Plot the points and draw a line through them. If all three points are not on the same line, double-check our calculations. Determining Whether a Point Lies on a Line The coordinates of an point on a line make the equation of that line true. The coordinates of an point not on a line make the equation of that line false. Eample Determine whether each point is on the line = + 6. A (, ) B ( 8.5,.5) Solution Substitute the - and -coordinates into the equation. = + 6 = ( 8.5) =.5 The point (, ) is not on the line = + 6. (See the graph in Eample.) The point ( 8.5,.5) is on the line = + 6. TIP Strictl speaking, = + 6 is an equation, not a line. However, its graph is a line, and mathematicians often use phrases such as the line = + 6. GRAPHS OF LINES 5

7 Using Intercepts to Graph a Linear Equation Finding the points where a graph crosses the aes is often useful. When a graph is a diagonal line, it intersects each ais at no more than one point. DEFINITIONS An -intercept is the -coordinate of a point where a graph intersects the -ais. A -intercept is the -coordinate of a point where a graph intersects the -ais. Eample Find the - and -intercepts of the equation 5 = 0. Then graph the equation. Solution The point that contains the -intercept has a -coordinate of zero, and the point that contains the -intercept has an -coordinate of zero. To find the -intercept, To find the -intercept, substitute 0 for in the equation substitute 0 for in the equation and solve for. and solve for. 5 = 0 5 = = = 0 = 0 = 0 = 0 = The -intercept is 0, so the point (0, 0) is on the line. The -intercept is, so the point (0, ) is on the line. Plot the points (0, 0) and (0, ), and then draw the line through them. REMEMBER Ever point on the -ais has -coordinate zero. Ever point on the -ais has -coordinate zero (0, 0) = 0 (0, ) 6 TIP To check our work, find a third ordered pair solution to the equation and make sure the graph of that ordered pair is a point on the line. 5 UNIT LINEAR EQUATIONS AND SYSTEMS

8 Simplifing and Graphing Linear Equations in One Variable Some linear equations have just one variable. These special linear equations have graphs that are vertical or horizontal lines. Eample A Simplif and graph = 0. B Simplif and graph + 5 =. Solution Divide both sides Solution Subtract 5 from both b to get = 5. To graph sides to get =. To graph the equation, plot some points the equation, plot some points that that have an -coordinate of 5, have a -coordinate of, such such as (5, ), (5, ), and as (, ), (0, ), and (, ). (5, ). The graph is shown The graph is shown below. below. = 5 (5, ) (5, ) (5, ) An equation that can be written in the form = k, where k is a constant, has a graph that is a vertical line. 5 (, ) (0, ) (, ) = An equation that can be written in the form = k, where k is a constant, has a graph that is a horizontal line. Calculating Slope from a Graph Another useful skill is knowing how to describe the slant of a line: which direction it runs and how steep it is. You can determine this information b finding a line s slope. DEFINITION The slope of a line is the ratio of its vertical change to its horizontal change. TIP Slope is sometimes described as rise over run. GRAPHS OF LINES 5

9 Eample 5 What is the slope of the line graphed at the right? Solution Pick an two points on the line, for eample (8, ) and (, ). Move from one point to the other b moving onl verticall and horizontall. To move from (8, ) to (, ), ou can move up units and then left 9 units. Therefore, the vertical change is +, and the horizontal change is 9. The slope of the line is + 9, which simplifies to. (, ) (8, ) Calculating Slope from a Linear Equation The vertical change between an two points is the difference in the -coordinates, and the horizontal change is the difference in the -coordinates. So if ou know the coordinates of two points on a line, ou can calculate the slope without a graph. DEFINITION The slope formula gives the slope of the line through two distinct points, (, ) and (, ). slope = rise run = vertical change horizontal change = Eample 6 Calculate the slope of the line with equation + =. Solution Find an two points on the line. In this case, it is eas to find the two points that contain the intercepts. Substitute 0 for : Substitute 0 for : 0 + = = = = 8. So the point (0, 6) is on the line. So the point ( 8, 0) is on the line. Then substitute the coordinates of the two points into the slope formula. Let (, ) = (0, 6) and (, ) = ( 8, 0). THINK ABOUT IT It doesn t matter which point ou use as (, ) or (, ). If ou reverse the substitution of (0, 6) and ( 8, 0), the signs change in the numerator and denominator, but ou still get the same slope ( 8) = 6 8 = slope = 0 6 = ( 8) 0 = 8 6 = The slope of the line with equation + = is. 5 UNIT LINEAR EQUATIONS AND SYSTEMS

10 Classification of Lines b Slope slope = + = + slope = = + A line with positive slope rises left to right. A line with negative slope falls left to right. zero vertical change 5 + zero horizontal change slope = 0 = 0 + slope = 0 = undefined A line with zero slope is horizontal. Problem Set A line with undefined slope is vertical. Find the - and -intercepts.. + =. + 5 = =. = 9. = = 5 Determine whether each point lies on the graph of =. 7. (5, ) 8. (, 7) 9. (, 9) GRAPHS OF LINES 55 Copright 009 K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations, without the epress prior written consent of K Inc.

11 Determine the slope of the line. 0. = +. = 7. = 6 8. =. = = Graph the equation = 0 9. = =. + =. = = + 7. = 6 5. = 6. + = Determine whether the slope of the line is positive, negative, zero, or undefined. 7. = = = = 0 Solve. *. Challenge Find a so the slope of the line through (, ) and (, a ) is. *. Challenge Find b so the slope of the line through (0.5b, 5) and (6, ) is. *. Challenge The slope of the line that passes through (, c) and (, ) is equal to the slope of the line that passes through (, ) and (0, ). What is the value of c? 56 UNIT LINEAR EQUATIONS AND SYSTEMS

12 Forms of Linear Equations You can write linear equations in different forms. All linear equations have the following characteristics in common: There is no variable with an eponent other than or zero. The variables are not multiplied together. It is possible to write the equation without an variable in the denominator of a fraction. The graph of the equation in a coordinate plane is a line. Converting Between Forms of Linear Equations Three forms of linear equations are most common. DEFINITIONS The standard form of a linear equation is A + B = C, where A, B, and C are integers, and A and B are not both zero. The slope-intercept form of a linear equation is = m + b, where m is the slope and b is the -intercept of the corresponding line. The point-slope form of a linear equation is = m( ), where m is the slope and (, ) is a point on the corresponding line. Eample A Write the equation + = 5 in slope-intercept form. Solution To convert to slope-intercept form, solve the equation for. + = 5 = + 5 Original equation Subtract from both sides. = + 5 Divide both sides b. B Write the equation = 5 ( + 7) in standard form. Solution To convert to standard form, isolate the variables from the constants. = 5 ( + 7) Original equation 5 0 = + 7 Multipl both sides b 5 to clear the fraction = = 7 Subtract from both sides. Add 0 to both sides. TIP To divide + 5 b, divide each term b : + 5 = + 5 = + 5 THINK ABOUT IT The final equation in Part B, + 5 = 7, is in standard form. An equivalent equation, also in standard form, is 5 = 7. FORMS OF LINEAR EQUATIONS 57

13 C Write the equation + = ( ) in slope-intercept form. Solution To convert to slope-intercept form, isolate the variable. + = ( ) Original equation + = Distribute. = Subtract from both sides. Graphing a Linear Equation in Slope-Intercept Form Eample Graph =. Solution This equation is in the form = m + b. The slope is. = + ( ) The -intercept is, so the point (0, ) is on the line. Plot the -intercept at (0, ). From that point, count the vertical and horizontal change in the slope ratio to plot another point on the line. As a good graphing habit, do this several times, plotting several more points. Then draw a line through the points. + (0, ) = Graphing a Linear Equation in Standard Form Eample Graph 5 + = 6 using two different methods. Solution Method A Convert the equation to slope-intercept form. Then use the graphing method shown in Eample above. 5 + = 6 = + 6 Subtract 5 from both sides. = 5 + Divide through b. Plot the -intercept at ( 0, ). Then count, using the slope 5, to plot another point. Method B Find both intercepts and use them to draw the line. Find the -intercept: Find the -intercept: 5 + = = = = 6 5 = 6 = 6 = 6 5 = Plot the point ( 5 6, 0 ). Plot the point ( 0, ). 0, ( ) Draw the line through these points. The line is the same as in Method A. + (., 0) = 6 (,.5) 58 UNIT LINEAR EQUATIONS AND SYSTEMS

14 Graphing a Linear Equation in Point-Slope Form Eample Graph = ( + ). Solution The slope is and (, ) is a point on the corresponding line. Plot (, ). Then count, using the slope, to plot another point. Another point on the line is (, 6). Draw the line through the two points (, 6) 6 5 (, ) 5 Linear Graph Famil: = m + b An changes to the parameters m and b in equations of the form f () = m + b will change the graphs of the lines. g() = f () = q() = h() = g() = + q() = + f () = h() = If ou change onl the parameter m, ou get lines through the origin, but with different slopes. If ou change onl the parameter b, ou get lines with the same slope but different -intercepts. FORMS OF LINEAR EQUATIONS 59

15 Problem Set Write each equation in the indicated form = 0 in slope-intercept form. = ( + ) in standard form. 0 + = 6 in slope-intercept form. + = in standard form Graph. 8. = = 8 0. =.. =.5 ( + ). + = ( + 6). = 0. = = 6. = ( 5) 7. + = 8. = 6. Solve. 9. Yosef is fencing his triangular backard. The equations of the three sides of fencing are 5 = 5, = + 7, and = 0. Graph the three equations that enclose Yosef s backard = ( ) in standard form 6. 5 = in slope-intercept form 7. = 9 ( + 7) in standard form = =. = = ( + 5) 6. 6 = ( ). 7 = 5. = = ( + ) 7. + = 8. = 0 * 0. Challenge Camilla is painting diagonal stripes on a lot to mark off parking spaces. She begins with two lines whose equations are = and = +. What are the equations of five other lines that mark the spaces? The lines must be parallel and equall spaced. Answer each question.. What effect does increasing a have on the graph of equation = a?. What effect does decreasing b have on the graph of equation = + b? 60 UNIT LINEAR EQUATIONS AND SYSTEMS

16 Writing Equations of Lines You can use given information to write the equation of a line. Finding a Linear Equation, Given the Slope and the -intercept You can write an equation of a line if ou know the slope and the -intercept. Eample Find an equation of the line with slope 5 and -intercept 6. Solution The slope-intercept form of a linear equation is = m + b, where m is the slope and b is the -intercept. So, in this case, m = 5 and b = 6. An equation of the line is = TIP You can convert an equation in slope-intercept form to standard form: = = 0 Finding a Linear Equation, Given the Slope and a Point on the Line Given the slope and a point on the line, ou can use the point-slope form of a linear equation to write an equation of the line. Eample Find an equation of the line that has slope and passes through the point (, ). Write the equation in slope-intercept form. Solution The point-slope form of a linear equation is = m( ), where m is the slope and (, ) is a point on the corresponding line. = m( ) Point-slope form ( ) = ( ) Substitute for m, for, and for. + = ( ) Simplif the left side of the equation. + = Distribute on the right side. = 5 Subtract from both sides. The equation of the line in slope-intercept form is = 5. Finding a Linear Equation, Given Two Points You do not need to see a graph in order to find an equation of a line. If ou know the coordinates of two points, or if ou know the coordinates of one point and the slope, ou have enough information to determine an equation. WRITING EQUATIONS OF LINES 6

17 Eample Find an equation of the line that passes through the points (, ) and (, ). Write the equation in standard form. Solution Use the known points on the line, (, ) = (, ) and (, ) = (, ), to calculate the slope m between them. m = = = = (Count on the grid from one point to the other to verif the slope.) Substitute the coordinates of one point and the slope into the point-slope form, and then convert the equation to standard form. = m( ) Point-slope form = ( ) Substitute for,, and m. = + Distribute. + = Isolate the variables from the constants. An equation of the line in standard form is + =. 5 THINK ABOUT IT In Eample, the point (, ) is substituted in the point-slope form for (, ). But ou can substitute an point that ou know is on the line. (, ) (, ) 5 Determining Whether Two Lines Are Parallel, Perpendicular, or Neither PROPERTIES OF PARALLEL AND PERPENDICULAR LINES Two nonvertical lines are parallel if and onl if the have equal slopes slope = slope = Two nonvertical lines are perpendicular if and onl if the have opposite reciprocal slopes slope = slope = THINK ABOUT IT Here is an alternate wa to state the propert of perpendicular lines: Two nonvertical lines are perpendicular if and onl if the product of their slopes is. Note that ( ) ( ) =. 6 UNIT LINEAR EQUATIONS AND SYSTEMS

18 Eample Determine whether each pair of lines is parallel, perpendicular, or neither. A + = = Solution Write the equation + = in slope-intercept form, = +. The slope is. The slope of = is. The slopes are not equal and are not opposite reciprocals. The lines are neither parallel nor perpendicular. B = + 5 = 6 6 Solution Write the equation = 6 6 in slope-intercept form, = +. The slopes of the lines are equal. The lines are parallel. C + = 7 = 5 Solution Write each equation in slope-intercept form. = + 7 = 5 The slopes are opposite reciprocals. The lines are perpendicular. Using Properties of Parallel or Perpendicular Lines to Find a Linear Equation Eample 5 A Find an equation of the line that is parallel to = 5 6 and passes through the point (, ). Write the equation in slope-intercept form. Solution The slope of the line = 5 6 is 5. Because parallel lines have equal slopes, the desired line also has a slope of 5. Substitute the slope and the coordinates of the known point into the slopeintercept form, and solve to find the desired -intercept. = m + b = 5 + b = 0 + b = b Use m and b to write the equation: = 5. B Find an equation of the line that is perpendicular to + = 0 and passes through the point ( 6, ). Write the equation in slopeintercept form. Solution Write the given equation in slope-intercept form: = +0. The slope is. Perpendicular lines have opposite reciprocal slopes, so the desired line has a slope of. Substitute the coordinates of the slope and the known point into the point-slope form, and then convert it to slopeintercept form. WRITING EQUATIONS OF LINES 6

19 = m( ) = _ [ ( 6)] = + = + The equation is = +. Problem Set Write an equation for each line.. m =, -intercept. m =, passes through the point (, ). m =, -intercept 0. m = 7, passes through the point (, ) 5. m =, passes through the point (7, 6) 6. m =, passes through the point (0, 9) 5 7. m = 0, -intercept 8. m =, passes through the point (, 7) 9. m = 5, -intercept 7 Write an equation for each line. 0. parallel to = 0 and passes through the point (, ). perpendicular to = 9 and passes through the point (, ). perpendicular to + = 5 and passes through the point ( 8, ) For each pair of points, do the following: A. Write an equation of the line in point-slope form. B. Write the equation in standard form. C. Write the equation in slope-intercept form. 7. (, 0) and (, ) 8. (, ) and (0, ) 9. (, 8) and (6, 5). undefined slope and passes through the point (, 6). perpendicular to 5 = 0 and passes through the point (, 6) 5. undefined slope and -intercept 7 6. parallel to = + and passes through the point (, ) 0. (, ) and (, 5). (8., ) and (, 7.6). (, ) and (, 8) Determine whether the lines are parallel, perpendicular, or neither.. + = 5 and = +. = + 0 and 5 = = and 7 = 0 6. = + and = + 7. = and = 8. 8 = 7 and 8 = 6 UNIT LINEAR EQUATIONS AND SYSTEMS

20 Solve. * 9. Challenge Sam wants to build a pool in an enclosed area behind his house. He wants one end of the pool to be perpendicular to the back wall of his house. If the line running along the house has equation = 0, and the point (, ) will rest on the end of the pool, find the equation of the line that is perpendicular to the back wall of Sam s house and passes through (, ). * 0. Challenge A decorator is painting a wall with stripes. The equation of the first gra stripe is = 7. Find the equation of the third stripe if it is parallel to the first and passes through the point (, ). WRITING EQUATIONS OF LINES 65

21 Applications: Linear Equations Man real-world situations involve slope and linear equations. Using a Linear Model to Estimate Eample Dan and Ben begin a road trip. The set the trip meter to 0. Dan drives first, and then Ben begins driving at noon. Ben drives at a fairl constant speed. At p.m., the trip meter reading is 80 miles. At :0 p.m., the reading is 67 miles. A Write a linear equation in slope-intercept form that approimates the relationship between the trip meter reading and the amount of time Ben drives. Graph the equation. B Describe what the slope represents. C Estimate the number of miles Dan drove. Solution A Slope-intercept form is = m + b. Let represent the number of hours Ben drives, and let represent the trip meter reading, in miles. At p.m., Ben has been driving for hours, and at :0 p.m., he has been driving for.5 hours, so ou know the following two data points: (, ) = (, 80) and (, ) = (.5, 67). Plot these two points. Find the slope: slope = m = = 67 mi 80 mi =.5 h h 87 mi.5 h = 58 mi = 58 mph h To solve for the -intercept b, substitute 58 for m and (, 80) for (, ) in the slope-intercept form: = m + b 80 = 58 + b 80 = 6 + b 6 = b The equation is = The graph of the equation is the line through the points (, 80) and (.5, 67). B Use units to figure this out. Since the numerator is miles and the denominator is hours, the ratio is miles per hour. In this situation, it is 58 miles Ben s average speed,. hour Miles THINK ABOUT IT If Ben drives at a constant speed, then the equation developed below is an eact model of the situation. But because Ben drives at a fairl constant speed, the equation is onl an approimate model. The onl eact data values that are known are those stated in the problem (, 80) (.5, 67) Hours 66 UNIT LINEAR EQUATIONS AND SYSTEMS

22 C When Ben begins driving, = 0. If = 0 in the equation = , then = 6. So, based on the facts presented in the problem, 6 miles is the best estimate of the number of miles Dan drove. Verif that 6 is reasonable for the -intercept of the graph. Using a Linear Model to Solve a Problem Eample Leslie earns a base pa of $.50 per hour. She also earns time-and-a-half for overtime (an hours over 0 hours in a week). A Write and graph a linear equation that models Leslie s earnings for a week in which she works at least 0 hours. B What does the slope of the equation represent? C Leslie reports her hours in quarter-hour increments. How man hours does Leslie need to work to earn $750 in one week? Solution A Let t represent the number of hours of overtime Leslie works in a week. Let e represent her earnings for that week, in dollars. 0 hours at Multipl $.50 b t is the number of $.50 per hour.5 for time-and-a-half overtime hours e = t e = t e = 8.75t The equation e = 8.75t models Leslie s earnings for a week in which she works at least 0 hours. To graph the equation, start with some ordered pairs and then draw a line through the points. t e Leslie works at least 0 hours, so onl points with values t 0 appl in this case. 000 B The slope, which is 8.75, represents the number of dollars per hour that Leslie earns at the overtime rate. C To find the number of hours Leslie needs to work to earn $750 in one week, substitute 750 for e and solve for t. e = 8.75t = 8.75t = 8.75t. = t Amount earned ($) If Leslie needs to works. overtime hours, she needs to work a total of 0 +. = 5. hours. Leslie reports her hours in quarter-hour increments, and 5. _ is between 5.5 and 5.5. So Leslie needs to work 5.5 hours to earn $750 in one week. e (5, 59.75) (0, 687.5) Overtime hours t APPLICATIONS: LINEAR EQUATIONS 67

23 Using a Linear Model to Interpret Solutions Eample Ed is a lumber dealer. He sells two tpes of decking, earning 0% profit on snthetic decking and 6% profit on wood decking. Ed has a goal of earning a $000 profit during the net month. A Write an equation that represents the sales, in dollars, of each tpe of decking Ed can sell to earn a $000 profit. B Graph the equation. C Identif three solutions to the equation and interpret their meaning. Solution A Let s represent sales, in dollars, of snthetic decking. Let w represent sales, in dollars, of wood decking. Write an equation: profit from sales of snthetic decking + profit from sales of wood decking = profit goal THINK ABOUT IT In this situation, either variable can be the independent variable. You could label the horizontal ais w and the vertical ais s, obtaining a different graph. 0.0s + 0.6w = 000 B Find the intercepts and use them to graph the equation. Label the horizontal ais s and the vertical ais w; points on the graph will have the form (s, w). Find the s-intercept: Find the w-intercept: 0.0s + 0.6w = s + 0.6w = s = w = s = w = 000 s = 0,000 w =,500 Plot the point (0,000, 0). Plot the point (0,,500). Connect the points with a line segment. The graph of the equation is a line, but in this application, onl positive values make sense. So just use Quadrant I. C Two solutions to the equation were found in Part B: (0,000, 0) and (0,,500). To find another solution, substitute a convenient value for one variable in the equation and solve for the other variable. Substitute 6000 for s and solve to find that w = So a third solution is (6000, 5000). Each solution is a combination of sales that result in a total profit of $ s + 0.6w = w = w = w = 800 w = 5000 Sales of wood decking ($) w,000 (0,,500) (0,000, 0) ,000 Sales of snthetic decking ($) s Solution What It Represents (0,000, 0) $0,000 sales of snthetic, $0 sales of wood (0,,500) $0 sales of snthetic, $,500 sales of wood (6000, 5000) $6000 sales of snthetic, $5000 sales of wood 68 UNIT LINEAR EQUATIONS AND SYSTEMS

24 Using a Linear Model to Make a Prediction Eample The table at the right gives the median weekl earnings of full-time workers in the United States for the ears Find a model for the data. Then use our model to predict the median weekl earnings in 00. Solution Let represent the ear and let represent the median weekl earnings in dollars. Graph the ordered pairs. (The graph is called a scatter plot.) The data points form a linear pattern, so a linear function is a good model for the data. Draw a line that approimates the pattern. Median weekl earnings ($) Year Year Median Weekl Earnings ($) Finding the line of best fit is a strateg that makes it eas to find an equation to model the data. This line of fit passes through (988, 85) and (00, 68). Find an equation of the line. Use the point-slope form of a linear equation slope = = m( ) 85 = 5.8( 988) 85 = 5.8,0. = 5.8,05. So a good linear model for the data is f () = 5.8,05.. To predict the median weekl earnings in 00, substitute 00 for. = 5.8,05. = ,05. = 7.6 According to this linear model, the predicted 00 median weekl earnings for a full-time worker is about $7. APPLICATIONS: LINEAR EQUATIONS 69

25 Problem Set For each problem, do the following: A. Define variables for the unknowns. B. Write an equation to model the problem. C. Graph the equation. D. Answer the question.. A child completes tasks to earn rewards. Yesterda, the child completed tasks and earned rewards. Toda, the child earned rewards b completing 6 tasks. How man tasks must the child complete to earn 0 rewards?. There is a linear relationship between a car s weight and its gas mileage. An average car that weighs 00 pounds gets 5 miles to the gallon, while a 000-pound car gets 7 miles per gallon. If a car gets 5 miles per gallon, how much should it weigh? For each problem, do the following: A. Write an equation to model the problem. B. Eplain what the slope represents. C. Answer the question.. A store sells cards b the bo. Engraved cards sell for $0 per bo, and all-occasion cards sell for $0 per bo. All purchases must total $50. If a person bought boes of engraved cards, how man all-occasion cards must he or she bu? 5. Gabrielle ran a marathon. From the start to the first checkpoint, Gabrielle ran at a constant rate of 0 miles in 0 minutes. At the second checkpoint, her total distance was 0 miles and her total time was 60 minutes. If the finish line is at 6. miles, what would be her time if she continued at the same rate? 6. A pot of water is heating on a stove at a constant rate. At 8 a.m., the temperature of the water is 8ºF. At 8:0 a.m., the temperature has risen to 00ºF. Find the temperature of the water at 8:08 a.m. 7. A carpenter, working at a constant rate, made 6 tables in das last week. This week, he made 9 tables in das. Find the number of tables the carpenter can make in 0 das.. A class has collected $0 to spend on a holida part. Each paper decoration costs $, and each part favor costs $.50. If the class bus 8 decorations, how man part favors can the class afford? 8. A doctor measured a bab s length at inches when the bab was one month old. At the bab s si-month doctor s appointment, the bab s length was 6 inches. Find the bab s length when the bab is one ear old if his length increases at a constant rate. 9. On September 5, the value of a stock was $ per share. Five das later, the stock s value was $00 per share. What is the value of the stock on October if the value changes at a constant rate? 0. Jamal weighed 00 pounds on the first of the month. Two weeks later, he weighed 96 pounds. What will Jamal weigh four weeks after his initial weigh-in if his weight decreases at a constant rate?. Trone earns a base pa of $0.75 per hour. He also earns time-and-a-half for overtime (an hours over 0 hours in a week). How man hours does Trone need to work to earn $500 in one week?. The average winning speed of a car race increased at a constant rate each ear the race was held. In 000, the winning speed was 0 miles per hour, then in 008 the winning speed was 0 miles per hour. Estimate the winning speed for UNIT LINEAR EQUATIONS AND SYSTEMS

26 . A phone compan charges a flat fee of $00 for one phone installation plus $0 for each additional phone jack. How man phones could be installed for $0?. A restaurant suggests that all patrons tip the help staff. A waitress earned a $.75 tip for a $5.0 meal and $.95 for a meal that cost $.50. Estimate the cost of a meal that will ield a $6 tip. For each problem, do the following: A. Define variables for the unknowns. B. Write an equation to model the data. C. Use our model to make the indicated prediction. 5. The table shows the average hours per da a person in the United States spent engaged in his or her children s education for the ears Predict the average hours per da a person will spend on his or her children s education in The table gives the emploment population percentages of 5-ear-olds to 7-ear-olds for the ears 98 to 989. Predict the percentage of 5-ear-olds to 7-ear-olds emploed in The table gives the average percentage of Americans who go to work each da. The data were collected for the ears Predict the percentage of Americans who will go to work each da in 00. Year Average Hours Year Percentage Year Percentage APPLICATIONS: LINEAR EQUATIONS 7

27 8. The table gives the primar energ consumption (trillion Btu) in the United States residential sector, beginning with the ear 999. Predict the primar residential energ consumption in The table gives the percentage of all individual ta returns filed in the United States for the ears for tapaers with adjusted gross income between $00,000 and $00,000. Predict the ear the percentage of filed income ta returns reaches 9 percent. Year Btu Year % UNIT LINEAR EQUATIONS AND SYSTEMS

28 Sstems of Linear Equations When equations with the same variables are grouped together, a sstem of equations is formed. Determining Whether a Given Ordered Pair Is a Solution to a Sstem DEFINITION A sstem of equations is two or more equations that contain the same variables. A given ordered pair is a solution to a sstem if it satisfies all of the equations. Eample Determine whether each point is a solution to the sstem: = 8 = 7 A (, ) B (8, ) Solution Substitute the coordinates of each point into each equation. A = 8 = 7 B = 8 = 7 ( ) = 8 = 8 = 8 7 The point (, ) is a solution to the sstem The point (8, ) is not a solution to the sstem because because it makes both equations true. it does not make both equations true. Solving Sstems b Graphing One wa to solve a sstem of equations is to graph each equation to determine where the intersect. The point of intersection represents the solution to the sstem. Eample Solve each sstem b graphing. = A = + 5 SYSTEMS OF LINEAR EQUATIONS 7

29 Solution Graph the two lines. The lines appear to intersect at (, ). You can check that (, ) is a solution to both equations. = = = = 0 B = = 5 = = (, ) 5 Solution Find the - and -intercepts of the first equation: 0 = 0 = 0 0 = 0 0 = = 0 = = 6 =. = 5 0 = 5 6 Then graph each line. Because the lines are parallel, the never intersect. This sstem has no solution Classifing Sstems of Linear Equations A sstem of two linear equations can have one solution, no solution, or infinitel man solutions. The table below shows how linear sstems are classified according to the number of solutions. Classifing Sstems of Linear Equations A linear sstem is consistent and independent if its graphs intersect at one point (one solution). A linear sstem is inconsistent if the lines are parallel (no solution). A linear sstem is consistent and dependent if the lines coincide (infinitel man solutions). 7 UNIT LINEAR EQUATIONS AND SYSTEMS

30 Solving Linear Sstems b the Substitution Method Algebraic methods, such as the substitution method, can be used to solve linear sstems and often ield more accurate results than graphing. HOW TO SOLVE A SYSTEM BY THE SUBSTITUTION METHOD Step Step Step Solve one equation for one of the variables. Substitute the epression obtained from Step in place of the corresponding variable in the other equation, and solve the resulting equation. Substitute the value of the variable from Step into either of the original equations. Solve this equation to find the value of the other variable. TIP You can alwas check our work b making sure that our solution satisfies both equations in the sstem. Eample Solve the sstem b substitution. + = 6 = 8 Solution Step Both equations are in standard form, so solve one of the equations for or. It is easiest to solve the first equation for. + = 6 = + 6 = 6 Step Substitute the epression 6 for in the second equation. The equation 8 = 8 is true for all real numbers. = 8 This means that there are infinitel man ( 6) = 8 solutions. 8 = 8 8 = 8 Step Because there is no single value of that is a solution, ou don t have to complete Step. So the sstem is consistent and dependent, and there are infinitel man solutions. TIP When a sstem solved algebraicall results in an equation that is true for all real numbers, the sstem has infinitel man solutions. When it results in an equation that is not true for an real number, the sstem has no solution. SYSTEMS OF LINEAR EQUATIONS 75

31 Solving Linear Sstems b the Linear Combination Method Another algebraic method to solve a sstem is the linear combination method. HOW TO SOLVE A SYSTEM BY THE LINEAR COMBINATION METHOD Step Step Step Step Convert both equations to standard form. Multipl one or both of the equations b a real number so that the coefficient of one variable in one equation is the additive inverse of the coefficient of the same variable in the other equation. Add the equations, and then solve the resulting equation. Substitute the value of the variable from Step into either of the original equations. Solve this equation for the second variable. REMEMBER The sum of a number and its additive inverse is 0. Eample Solve the sstem b linear combination. + 5 = = Solution Step Convert the second equation to standard form + 5 = b mutipling through b and subtracting from + = both sides. Step Multipl the first equation b and the second equation b so that the coefficients of are additive inverses. Step Add the equations to eliminate. Solve the result for = = 6 = = Step Substitute for in either of the original equations to find. + 5 ( ) = 5 = = 6 = So the solution to the sstem is (, ). THINK ABOUT IT The solution to a linear sstem of equations does not change when ou multipl one or both of the equations b a nonzero real number. 76 UNIT LINEAR EQUATIONS AND SYSTEMS

32 Solving a Sstem of Three Equations with Three Variables It is also possible to solve a sstem of three equations containing three variables. The solution of a sstem of three equations in three variables can have one solution, infinitel man solutions, or no solution. A sstem of three equations is solved b finding ordered triples that satisf all of the equations. NOTATION The solution of a sstem of three equations in three variables,, z is called an ordered triple and is written (,, z). HOW TO SOLVE A SYSTEM OF THREE EQUATIONS WITH THREE VARIABLES Step Step Step Step Step 5 Convert each equation to standard form. Eliminate one variable from two of the equations. Using these two equations and substitution or linear combination, solve for one of the remaining variables. Substitute the value of the variable from Step into either of the equations in Step. Solve this equation for the second variable. Substitute the value of the variables from Step and Step into an of the original equations. Eample 5 Solve the sstem. + 5 z = z = + z = Solution Step The sstem has three equations in standard form. Eliminate the z variable from the first two equations b combining them. Step Multipl the second equation b. Then combine the second equation with the third equation to eliminate the z variable. Step Now ou have a sstem of two linear equations with two variables. Combine the equations to solve the sstem. + 5 z = z = + 6 = z = 6 + z = = = = 5 = 6 = Step Use the value of to solve for in an of the two equations. Then substitute the value of and into one of the original equations to solve for z. + 6 ( ) = = = + 5 ( ) z = 8 0 z = 8 8 z = 8 z = 0 The solution to the sstem is (,, 0). TIP Check that our solution satisfies ever equation in the sstem. SYSTEMS OF LINEAR EQUATIONS 77

33 Problem Set Determine whether the given ordered pair or triple is a solution to the sstem.. (, ), = =. (, 0), = 6 = (5, 7), = 6 + = 9. (, ), 6 = 7 = 5. (, ), 5 6 = 6 5 = 6. (, 0, ), + z = + + z = z = Solve each sstem b graphing... = + 6 = = + = + Solve each sstem b substitution = 8 = + 6 = + = Solve each sstem b linear combination = 7 + = 7 = 9 + = = = 5 = 6 + =..8 = =.9 + = = + 7. (,, ), 8. (,, ), 9. (0,, ), 0. (0,, 5), = = 0 + = = 9 = + + z = z = z = + 7 z = + z = z = + z = 5 + z = + = + + z = + z = z = = 9 + = = z = z = + z = 5 + z = + z = 5 + 6z = = 9 = z = 8z = 0 + = 78 UNIT LINEAR EQUATIONS AND SYSTEMS

34 Applications: Linear Sstems Man real-world problems can be solved with a sstem of linear equations. Solving an Investment Problem Eample Linda invested $000, separating it into two mutual funds. The percent return on Star Spangled Corporation at the end of a ear was 5%, and the percent return on Joe Dow s Industries was 7%. The combined return on both funds was $56. How much did Linda invest in each fund? Solution Step Let represent the dollar amount invested in Star Spangled Corporation. Let represent the dollar amount invested in Joe Dow s Industries. Write two equations to represent the situation. Equation : amount invested in Star Spangled Corp. + amount invested in Joe Dow s Industries = total amount invested TIP Percent means hundredths. To write a percent in decimal form, use the meaning of percent: 5% = 5 00 = 0.05 Or just remove the % smbol and move the decimal point two places to the left: 5% = 5.% = Equation : + = 000 amount earned in Star Spangled Corp. + amount earned in Joe Dow s Industries = total amount earned Step = 56 Solve the sstem. + = 000 Use Equations and to write a sstem = = 00 Multipl both sides of Equation b 0.05 to get = 56 opposite -coefficients. 0.0 = 56 Add to eliminate. = 800 Solve for = 000 Substitute 800 for in Equation. = 00 Solve for. Linda invested $00 in Star Spangled Corporation and $800 in Joe Dow s Industries. APPLICATIONS: LINEAR SYSTEMS 79

35 Solving a Miture Problem Eample A certain brand of orange juice has 6.7% of the recommended dail allowance (RDA) of vitamin C per ounce. The same brand of pineapple juice has 5.5% of the RDA of vitamin C per ounce. An 8-ounce miture of these juices contains % of the RDA of vitamin C. How much orange juice and how much pineapple juice are in the miture? Solution Step Let represent the number of ounces of orange juice. Let represent the number of ounces of pineapple juice. Write two equations. Equation : The first equation represents the sum of the two amounts of juice, which must equal the total amount of the miture, in ounces. + = 8 Equation : The second equation represents portions of the RDA of vitamin C. One ounce of orange juice contains 0.67 of the RDA of vitamin C, so ounces contain 0.67 of the RDA of vitamin C. One ounce of pineapple juice contains of the RDA of vitamin C, so ounces contain of the RDA of vitamin C. The miture contains 0. of the RDA of vitamin C, so 8 ounces contain 0.(8), or 0.96 of the RDA of vitamin C = 0.96 Step Solve the sstem. + = = = = = = 960 = 50 Add to eliminate..6 Divide b to solve for Substitute.6 for in Equation.. Solve for. Multipl the second equation b 000 to eliminate the decimals. Multipl both sides of Equation b 5 to get opposite -coefficients. There are about.6 ounces of orange juice and about. ounces of pineapple juice in the miture. TIP Percent is a ratio that compares a number to 00. Miture problems can use ratios in other forms, such as unit prices or servings per container. Think about it You could write Equation, using percents instead of their decimal equivalents, as follows: = (8) The sstem would have the same solution. 80 Unit linear equations and sstems

36 Solving a Sports Problem Eample In the National Basketball Association season, Cleveland s LeBron James scored 50 points, making a total of shots, including one-point free throws, two-point field goals, and three-point field goals. He made 9 more two-point field goals than three-point field goals and one-point free throws combined. Find the number of each tpe of shot he made. Solution Step Let represent the number of three-point field goals made, let represent the number of two-point field goals made, and let z represent the number of one-point free throws made. Write three equations. Equation : Number of shots made: + + z = Equation : Number of points scored: + + z = 50 Equation : Relationship between tpes of shots: = + z z = 9 Step Solve the sstem. + + z = + + z = 50 + z = z = + z = 9 = 6 = z = z = 50 + z = z = 50 + z = 66 + z = 888 = 6 = z = 79 + z = z = 59 Eliminate and z from the first equation and the third equation b adding them. Substitute 68 for in Equation and Equation. Simplif. Solve the resulting sstem. Substitute 68 for and for in Equation and solve for z. LeBron James made three-point field goals, 68 two-point field goals, and 59 one-point free throws. APPLICATIONS: LINEAR SYSTEMS 8

37 Problem Set For each problem, do the following: A. Define variables for the unknowns. B. Write a sstem of equations to model the problem. C. Solve the sstem. D. Give our answer in a complete sentence.. Tler invested $500 in certificates of deposit. One certificate of deposit pas % in interest, and the other pas 6.5% in interest. He earned $0 combined from the certificates of deposit. How much did Tler invest in each certificate of deposit?. A grocer store sells pears for $.99 per pound and apples for $.9 per pound. A customer purchases 9 pounds of pears and apples for $5.. How man pounds of each did the customer purchase?. At a movie theater, an adult ticket costs $0 and a child ticket costs $6. There were 50 people at a movie showing. The revenue for that showing was $9. How man adults were at the showing? How man children were at the showing?. A football team scored on 7 plas in a game for a total of points. How man touchdowns did the team score? How man field goals did the team make? (Count touchdowns as 7 points and field goals as points.) 5. Kianna invested $6000, separating it into two mutual funds. The percent return on Tidewater Corporation at the end of a ear was %, and the percent return on Lincoln, Inc., was %. The combined return on both funds was $5. How much did Kianna invest in each fund? 6. A chemistr eperiment requires a miture of a 0% hdrogen solution and a 0% hdrogen solution. How man liters of each solution are required for liters of % hdrogen solution? 7. Chang and his brother, Min, use different cell phone plans from the Global Connection phone compan. In one month, the two brothers use a total of 500 minutes. Chang pas $0.0 per minute used and Min pas $0.5 per minute used. If their total bill adds up to $, how man minutes did each brother use for the month? 8. Darion spent $75 on a coat and three shirts. Each shirt cost the same amount. The coat cost four times the amount of one shirt. What was the price of the coat? What was the price of each shirt? 9. The Perfect Electronics store bus computers from two different companies. Compan A produces defective computers for ever 00 computers sold, and Compan B produces 5 defective computers for ever 50 computers sold. In one ear, the store purchases a total of 000 computers. If the total number of defective computers for that ear was 8, how man computers did the store purchase from each compan? 0. Mario invested $700 in two mutual fund accounts. One account pas 7.5% in interest, and the other pas 9% in interest. He earned $66 combined from the accounts. How much did Mario invest in each account?. A farmer harvests corn and wheat. He earns $ per bushel of corn and $.50 per bushel of wheat. Last ear, he harvested a total of 000 bushels and earned $ How man bushels of corn and wheat did the farmer harvest?. Sam s Chicken bus sets of small aprons and large aprons. A set of 5 small aprons costs $0, and a set of large aprons costs $0. The restaurant spends $80 on 7 sets of aprons. How man individual small aprons and individual large aprons did the purchase altogether?. At a coffee shop, a bagel costs $.9 and a muffin costs $.50. On Monda, sales for muffins and bagels totaled $6.50. There were 5 fewer bagels sold than muffins. How man bagels were sold? How man muffins were sold?. The residents of a cit voted on a proposal. There were 0,000 people who cast a vote. There were four times the number of people for the proposal than people against the proposal. How man people voted for the proposal? How man people voted against the proposal? 5. A total of $.50 worth of quarters and dimes was in a parking meter. If the total number of coins was 7, how man of each coin were in the meter? 8 Unit linear equations and sstems