Systems of Equations. from Campus to Careers Fashion Designer


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1 Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations. Solving Sstems of Equations in Three Variables. Problem Solving Using Sstems of Three Equations.6 Solving Sstems of Equations Using Matrices.7 Solving Sstems of Equations Using Determinants Chapter Summar and Review Chapter Test Cumulative Review Fashion designers help create the billions of clothing articles, shoes, and accessories purchased ever ear b consumers. Fashion design relies heavil on mathematical skills, including knowledge of lines, angles, curves, and measurement. Designers also use mathematics in the manufacturing and marketing parts of the industr as the calculate labor costs and determine the markups and markdowns involved in retail pricing. In Problem of Stud Set., we will eamine the production side of fashion design as we determine the number of coats, shirts, and slacks that can be made with the available labor. JOB TITLE: Fashion Designer EDUCATION: Man communit colleges and vocational training schools provide training for fashion industr jobs. JOB OUTLOOK: The best opportunities will be in designing clothing sold in department stores and retail chains. ANNUAL EARNINGS: $69,70 FOR MORE INFORMATION:
2 6 Chapter Sstems of Equations Objectives Determine whether an ordered pair is a solution of a sstem. Solve sstems of linear equations b graphing. Use graphing to identif inconsistent sstems and dependent equations. SECTION. Solving Sstems of Equations b Graphing The red line in the graph shows the cost for a compan to produce a given number of skateboards. The blue line shows the revenue the compan will earn for selling a given number of those skateboards. The graph offers the compan important financial information. The production costs eceed the revenue earned if fewer than 00 skateboards are sold. In this case, the compan loses mone. The revenue earned eceeds the production costs if more than 00 skateboards are sold. In this case, the compan makes a profit. Production costs equal revenue earned if eactl 00 skateboards are sold. This fact is indicated b the point of intersection of the two lines, (00, 0,000), which is called the breakeven point. Production costs and sales revenues $0,000 0,000 0,000 0,000 Here, cost eceeds revenue. Breakeven point: Cost equals revenue. (00, 0,000) Here, revenue eceeds cost. Revenue Cost Number of skateboards 800 From Chapter, we know that the lines in the graph that show the cost and the revenue can be modeled b linear equations in two variables. Together, such a set of equations is called a sstem of equations. In general, when two equations with the same variables are considered simultaneousl (at the same time), we sa that the form a sstem of equations. We will use a left brace { when writing a sstem of equations. An eample is e Read as the sstem of equations and. Determine whether an ordered pair is a solution of a sstem. A solution of a sstem of equations in two variables is an ordered pair that satisfies both equations of the sstem. Self Check Determine whether (6, ) is a solution of the sstem: 0 e es 0 Now Tr Problem 6 EXAMPLE Determine whether (, ) is a solution of each sstem of equations. a. e b. e Strateg We will substitute the  and coordinates of corresponding variables in both equations of the sstem. (, ) for the
3 . Solving Sstems of Equations b Graphing 7 WHY If both equations are satisfied (made true) b the  and coordinates, the ordered pair is a solution of the sstem. Solution a. To determine whether (, ) is a solution, we substitute for and for in each equation. Check: ( ) () 6 First equation. True Since (, ) satisfies both equations, it is a solution of the sstem. b. Again, we substitute for and for in each equation. Check: () ( ) First equation. True ( ) 9 Second equation. True Second equation. False Although (, ) satisfies the first equation, it does not satisf the second. Because it does not satisf both equations, (, ) is not a solution of the sstem. Teaching Eample Determine whether (, ) is a solution of the 8 sstem: e Answer: no Solve sstems of linear equations b graphing. To solve a sstem of equations means to find all the solutions of the sstem. One wa to solve a sstem of linear equations in two variables is to graph each equation and find where the graphs intersect. The Graphing Method. Graph each equation on the same rectangular coordinate sstem.. Find the coordinates of the point (or points) where the graphs intersect. These coordinates give the solution of the sstem.. If the graphs have no point in common, the sstem has no solution.. Check the proposed solution in both of the original equations. When a sstem of equations (as in Eample ) has at least one solution, the sstem is called a consistent sstem. EXAMPLE Solve the sstem b graphing: e Strateg We will graph both equations on the same coordinate sstem. WHY The graph of a linear equation is a picture of its solutions. If both equations are graphed on the same coordinate sstem, we can see whether the have an common solutions.
4 8 Chapter Sstems of Equations Self Check Solve the sstem b graphing: e (, ) = Now Tr Problem Teaching Eample Solve the sstem b graphing: e Answer: (, ) = + + = (, ) = (, ) Solution Although infinitel man ordered pairs (, ) satisf, and infinitel man ordered pairs (, ) satisf, onl the coordinates of the point where the graphs intersect satisf both equations. From the graph, it appears that the intersection point has coordinates (, ). To verif that it is the solution, we substitute for and for in both equations and verif that (, ) satisfies each one, as shown below. 0 (, ) (, 0) 0 (0, ) (, ) + = 0 (, ) 0 (, ), 0 (0, ) (, ) Use the intercept method to graph each line. The point of intersection gives the solution of the sstem. = Check: The first equation The second equation () () Since (, ) makes both equations true, it is the solution of the sstem. Success Tip Since accurac is crucial when using the graphing method to solve a sstem: Use graph paper. Use a sharp pencil. Use a straight edge. Use graphing to identif inconsistent sstems and dependent equations. When a sstem has no solution (as in Eample ), it is called an inconsistent sstem.
5 . Solving Sstems of Equations b Graphing 9 EXAMPLE 6 Solve the sstem e b graphing, if possible. 6 Strateg We will graph both equations on the same coordinate sstem. WHY The graph of a linear equation is a picture of it solutions. If both equations are graphed on the same coordinate sstem, we can see whether the have an common solutions. Solution Using the intercept method, we graph both equations on one set of coordinate aes, as shown below. Self Check 6 Solve the sstem e 6 b graphing, if possible. = 6 = 6 6 (, ) 0 (, 0) 0 (0, ) (, ) 6 (, ) 6 0 (6, 0) 0 (0, ) 6 (, 6) Now Tr Problem Self Check Answer no solutions Teaching Eample Solve the sstem u b graphing, if possible. Answer: no solutions = + 6 = + = = 6 In this eample, the graphs are parallel, because the slopes of the two lines are equal and the have different intercepts. We can see that the slope of each line is b writing each equation in slopeintercept form Since the graphs are parallel lines, the lines do not intersect, and the sstem does not have a solution. It is an inconsistent sstem. The solution set is the empt set, which is written. When the equations of a sstem of two equations in two variables have different graphs (as in Eamples and ), the equations are called independent equations. Two equations with the same graph are called dependent equations.
6 0 Chapter Sstems of Equations THINK IT THROUGH Bachelor s Degrees Women now earn more associate s, bachelor s and master s degrees than their male counterparts. Women also earned nearl half of the Ph.D.s as well as professional degrees, which include medical, law and dental degrees. CNN/Mone, April 7, 00 Eamine the graph below. Determine the point of intersection of the graph and eplain its importance. (Note: The ais does not need to be scaled for ou to answer this question.) (98, 0); in 98, 0% of the bachelor s degrees that were awarded went to men and 0% went to women. Percent of degrees Men 970 Women Academic ear ending Source: U.S. Department of Education, National Center for Education Statistics Self Check Solve the sstem b graphing: e = = Now Tr Problem Self Check Answer There are infinitel man solutions; three of them are (0, ), (, 0), and (, ). Teaching Eample Solve the sstem 0 b graphing: u Answer: There are infinitel man solutions; three of them are (, 0), (0, ), and (, ). = 0 = EXAMPLE Solve the sstem b graphing: u 8 Strateg We will graph both equations on the same coordinate sstem. WHY If both equations are graphed on the same coordinate sstem, we can see whether the have an common solutions. Solution We graph each equation on one set of coordinate aes, as shown below. Since the graphs coincide (are the same), the sstem has infinitel man solutions. An ordered pair (, ) that satisfies one equation also satisfies the other. From the graph we see that (, 0), (0, ), and (, ) are three of the infinitel man solutions. Because the two equations have the same graph, the are dependent equations. Graph b using the slope and intercept. m b Slope intercept: (0, ) Graph b using the intercept method. 8 0 (, ) (, 0) 0 (0, ) (, ) = = 6 6
7 . Solving Sstems of Equations b Graphing We now summarize the possibilities that can occur when two linear equations, each in two variables, are graphed. Solving a Sstem of Equations b the Graphing Method If the lines are different and intersect, the equations are independent, and the sstem is consistent. One solution eists. It is the point of intersection. If the lines are different and parallel, the equations are independent, and the sstem is inconsistent. No solution eists. If the lines coincide, the equations are dependent, and the sstem is consistent. Infinitel man solutions eist. An point on the line is a solution. If each equation in one sstem is equivalent to a corresponding equation in another sstem, the sstems are called equivalent. EXAMPLE Solve the sstem b graphing: Strateg We will use the multiplication propert of equalit to clear both equations of fractions and solve the resulting equivalent sstem b graphing. WHY It is usuall easier to solve sstems that do not contain fractions. Solution We multipl both sides of b to eliminate the fractions and obtain the equation. We multipl both sides of b to eliminate the fractions and obtain the equation 8. The original sstem μ The new sstem Multipl b Simplif a b a b μ a b () 8 Multipl b Simplif is equivalent to the original sstem and is easier to solve, since it has no fractions. If we graph each equation in the new sstem, it appears that the lines intersect at (, ). Verif that (, ) is the solution b substituting for and for in each equation of the original sstem. Self Check Solve the sstem b graphing: (, ) + = 9 = (, ) Now Tr Problem 7
8 Chapter Sstems of Equations Teaching Eample Solve the sstem b graphing: 6 Answer: (, ) (, ) = 9 = 0 0 (, ), 0 0, (, ) 8 0 (, ) (, 0) 0 8 (0, 8) 6 (, 6) = (, ) = 8 Caution! When checking a solution of a sstem of equations, alwas substitute the values of the variables into the original equations. Using Your CALCULATOR Solving Sstems b Graphing The graphing method has limitations. First, the method is limited to equations with two variables. Sstems with three or more variables cannot be solved graphicall. Second, it is often difficult to find eact solutions graphicall. However, the TRACE and ZOOM capabilities of graphing calculators enable us to get ver good approimations of such solutions. For eample, to solve the sstem e with a graphing calculator, we first solve each equation for so that we can enter the equations into the calculator. After solving for, we obtain the equivalent sstem: 6 If we use window settings of [ 0, 0] for and [ 0, 0] for, the graphs of the equations will look like those in figure (a) on the net page. If we zoom in on the intersection point of the two lines and trace, we will get an approimate solution like the one shown in figure (b). To get better results, we can do more zooms. We would then find that, to the nearest hundredth, the solution is (.6, 0.9). We can also find the intersection of the two lines b using the INTERSECT feature found under the CALC menu. After graphing the lines and using INTERSECT, we obtain the displa shown in figure (c), which shows the approimate coordinates of the point of intersection to be (.6, 0.9). (continued)
9 . Solving Sstems of Equations b Graphing (a) (b) (c) ANSWERS TO SELF CHECKS. es. (, ). no solutions = (, ) = 6 + = = 6. There are infinitel man. solutions; three of them are (0, ), (, 0), and (, ). = = (, ) = + = 9 (, ) SECTION. STUDY SET VOCABULARY Fill in the blanks.. e is called a sstem of linear equations.. When a sstem of equations has one or more solutions, it is called a consistent sstem.. If a sstem has no solutions, it is called an inconsistent sstem.. If two equations have different graphs, the are called independent equations.. Two equations with the same graph are called dependent equations. 6. When solving a sstem of two linear equations b the graphing method, we look for the point of intersection of the two lines. CONCEPTS 7. Refer to the illustration. Determine whether a true or a false statement would be obtained when the coordinates of a. point A are substituted into the equation for line l. true A l b. point B are substituted into the equation for C 6 line l. false c. point C are substituted into B l the equation for line l. true d. point C are substituted into the equation for line l. true Selected eercises available online at
10 Chapter Sstems of Equations 8. Refer to the illustration. a. How man ordered pairs satisf the equation? Name three. infinitel man; (, 6), (0, ), (, 0) b. How man ordered pairs satisf the equation? Name three. infinitel man; (, ), (0, ), (, ) c. How man ordered pairs satisf both equations? Name it or them. one; (0, ) + = = 9. a. The intercept method can be used to graph the equation 8. Complete the following table (, ) (, 0) (0, ) (, ) b. What is the intercept of the graph of 8? What is the intercept? (, 0), (0, ) 0. a. To graph, we can pick three numbers for and find the corresponding values of. Complete the following table. 0 (, ) (, ) (0, ) 7 (, 7) b. We can also graph if we know the slope and the intercept of the line. What are the? m, (0, ). Write a sstem of two linear equations that has a. onl one solution, (, ). e (answers ma var) b. an infinite number of solutions. e (answers ma var) 0 c. no solution. e (answers ma var). Estimate the solution of the sstem of linear equations shown in the displa below. Then check our answer. (, ) NOTATION Fill in the blanks.. A left brace is often used when writing a sstem of equations.. The solution of a sstem of two linear equations in two variables is written as an ordered pair. GUIDED PRACTICE Determine whether the ordered pair is a solution of the sstem of equations. See Eample. 0. (, ) ; u 6. (, ) ; es no 7. (, ) ; u 8. (, ) ; 0 no es Solve each sstem b graphing. See Eample e 0. e + = 6 (, ) = = e 9 e 6 + = (, ). e. e 7 7 (, ) + = = 7 (, ) = + = 7
11 . Solving Sstems of Equations b Graphing Solve each sstem b graphing. If a sstem is inconsistent or if the equations are dependent, so indicate. See Eamples. 6. e. e 0 no solution, inconsistent infinitel man solutions, sstem dependent equations TRY IT YOURSELF Solve each sstem b graphing. If a sstem is inconsistent or if the equations are dependent, so indicate e. e 0 6 = = + = 6 0 = + = 0 (0, 0) = 0 + = 0 = 6 (, ). e 6. e 6 7 = + = 6 infinitel man solutions, dependent equations = no solution, inconsistent sstem + = 7 Solve each sstem b graphing. See Eample = + = (, ) = + 6 infinitel man solutions, dependent equations + = 6 9. μ 0. μ 6 = 6 no solution, inconsistent sstem = 6 (, ) = + = 0. e. e 0 0 = = ( ) + = 0, + = (, ) + = 0 (0, 0) = 0. e 6. e 0 0 (, ) + = 0 + = 7 7. e 8. e = = + (, ) 6 = 7 = + (, )
12 6 Chapter Sstems of Equations 9. u 0. u = + (, ) = (, ) = =. e. e 9 = (, ) = + = 9 (, ) + =. μ. e 0 = (, ) + = 0 = = (, ). 6. e 8 = (, ) = (, ) = = 8 Use a graphing calculator to solve each sstem. Give all answers to the nearest hundredth e 0. e ( 0.7,.69) (.,.60) e. e ( 7.6, 7.0) (.,.) APPLICATIONS. MAPS See the following illustration. Name the cities that lie along Interstate 0. Name the cities that lie along Interstate. What cit lies on both interstate highwas? Gallup, Grants, Albuquerque, Tucumcari; Las Vegas, Santa Fe, Albuquerque, Socorro, Las Cruces; Albuquerque 6 Raton Farmington Las Vegas Santa Fe Grants Tucumcari Gallup 8 0 Albuquerque Clovis Vaughn Socorro 70 Lordsburg Roswell 8 Las Cruces Carlsbad. THE INTERNET The graph on the net page shows the growing importance of the Internet in the dail lives of Americans. Determine when the time spent on the following activities was the same. Approimatel how man hours per ear were spent on each? a. Internet and reading magazines 00, 0 hr b. Internet and reading newspapers 00, 8 hr c. Internet and reading books 000, 0 hr d. Reading newspapers and listening to recorded music 00, 00 hr and 007, 7 hr 6 7. u infinitel man solutions, dependent equations 8. e no solution, inconsistent sstem 6
13 . Solving Sstems of Equations b Graphing 7 Hours per ear per person Recorded music Dail Newspapers Books Video games Consumer Internet Consumer Magazines 7. SUPPLY AND DEMAND The demand function in the illustration describes the relationship between the price of a certain camera and the demand for the camera. a. The suppl function, S(), describes the relationship between the price of the camera and the number of cameras the manufacturer is willing to suppl. Graph this function. b. For what price will the suppl of cameras equal the demand? $0 c. As the price of the camera is increased, what happens to suppl and what happens to demand? Suppl increases and demand decreases. '99 '00 '0 '0 '0 '0 '0 '06 '07 Year Source: Veronis Suhler Stevenson. HEARING TESTS See the illustration below. At what frequenc and decibel level were the hearing test results the same for the left and right ear? Write our answer as an ordered pair. (,000, 0) Hearing level (decibels) Normal Hearing Range important for speech Right ear Left ear 0 00,000,000,000 8,000 Frequenc (ccles per second) 6. BUSINESS Estimate the breakeven point (where cost revenue) on the graph below. Then eplain wh is it called the breakeven point. (0,,00) Production costs and sales revenues (in dollars),000,000,000,000,000 0 Total cost Total revenue Number of widgets produced Number of cameras Demand function Price per camera ($) 8. COST AND REVENUE The function C() gives the cost for a college to offer sections of an introductor class in CPR (cardiopulmonar resuscitation). The function R() 80 gives the amount of revenue the college brings in when offering sections of CPR. a. Find the breakeven point (where cost revenue) b graphing each function on the same coordinate sstem. (,,00) b. How man sections does the college need to offer to make a profit on the CPR training course? more than 9. NAVIGATION The paths of two ships are tracked on the same coordinate sstem. One ship is following a path described b the equation 6, and the other is following a path described b the equation. a. Is there a possibilit of a collision? es b. What are the coordinates of the danger point? (.7, 0.) c. Is a collision a certaint? no
14 8 Chapter Sstems of Equations 60. AIR TRAFFIC CONTROL Two airplanes fling at the same altitude are tracked using the same coordinate sstem on a radar screen. One plane is following a path described b the equation, and the other is following a path described b the equation 7. Is there a possibilit of a collision? no WRITING 6. Suppose the solution of a sstem of two linear equations is, 8. Knowing this, eplain an drawbacks with solving the sstem b the graphing method. 6. Can a sstem of two linear equations have eactl two solutions? Wh or wh not? REVIEW Let and g() Find each value ƒ( ) 6. ƒ(0) g() g( 0) 67. Determine the domain and range of ƒ(). D: the set of real numbers, R: the set of all real numbers greater than or equal to 68. Find the slope of the line passing through the points (, 8) and (, 8) The area of the square on the right is 8 square centimeters. Find the area of the shaded triangle. 0. cm 70. If the area of the circle on the right is 9p square centimeters, find the area of the square. 96 cm Objectives Solve sstems of linear equations b substitution. Solve sstems of linear equations b the addition (elimination) method. Use substitution and addition (elimination) to identif inconsistent sstems and dependent equations. Determine the most efficient method to solve a linear sstem. SECTION. Solving Sstems of Equations Algebraicall The graphing method provides a wa to visualize the process of solving sstems of equations. However, it can sometimes be difficult to determine the eact coordinates of the point of intersection. We now discuss two other methods, called the substitution and the addition methods, that can be used to find the eact solutions of sstems of equations. Solve sstems of linear equations b substitution. To solve a sstem of two equations in two variables b the substitution method, we follow these steps. The Substitution Method. Solve one equation for a variable preferabl one with a coefficient of or. If this is alread done, go to step. We call the equation found in step the substitution equation.. Substitute the epression for or for obtained in step into the other equation and solve it.. Substitute the value of the variable found in step into the substitution equation to find the value of the remaining variable.. State the solution.. Check the proposed solution in both of the original equations. Write the solution as an ordered pair.
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