2-1: Solving Systems Of Equations In Two Variables

Madison is thinking about leasing a car fortwo years. The dealership says that they will lease her the car she hasp li c a tichosen for 326 per month with only 200 down. However, if shepays 1600 down, the lease payment drops to 226 per month. What is the breakeven point when comparing these lease options? Which 2-year lease should shechoose if the down payment is not a problem? This problem will be solved inExample 4.CONSUMER CHOICESonAp Solve systemsof equationsgraphically. Solve systemsof equationsalgebraically.l WorealdOBJECTIVESSolving Systems of Equationsin Two VariablesR2-1The break-even point is the point in time at which Madison has paid thesame total amount on each lease. After finding that point, you can more easilydetermine which of these arrangements would be a better deal. The break-evenpoint can be determined by solving a system of equations.A system of equations is a set of two or more equations. To “solve” a systemof equations means to find values for the variables in the equations, which makeall the equations true at the same time. One way to solve a system of equations isby graphing. The intersection of the graphs represents the point at which theequations have the same x-value and the same y-value. Thus, this ordered pairrepresents the solution common to both equations. This ordered pair is calledthe solution to the system of equations.ExampleGraphingCalculatorTipYou can estimate thesolution to a system ofequations by using theTRACE function onyour graphing calculator.1 Solve the system of equations by graphing.3x 2y 6x y 2( 2, 0)First rewrite each equation of the system inslope-intercept form by solving for y.3x 2y 6x y 2yy x 23becomesy x 32y x 2Oxy 23 x 3Since the two lines have different slopes, the graphs of the equations areintersecting lines. The solution to the system is ( 2, 0).As you saw in Example 1, when the graphs of two equations intersect there isa solution to the system of equations. However, you may recall that the graphs oftwo equations may be parallel lines or, in fact, the same line. Each of thesesituations has a different type of system of linear equations.A consistent system of equations has at least one solution. If there is exactlyone solution, the system is independent. If there are infinitely many solutions,the system is dependent. If there is no solution, the system is inconsistent. Byrewriting each equation of a system in slope-intercept form, you can more easilydetermine the type of system you have and what type of solution to expect.Lesson 2-1Solving Systems of Equations in Two Variables 67

The chart below summarizes the characteristics of these types of yyy2y 4x 14y 3x 23y 6x 21y x 1When graphsresult in lines thatare the same line,we say the linescoincide.Oy 0.4x 2.25OxxOxy 0.4x 3.1y 3x 2y x 12y 4x 14 y 2x 73y 6x 21 y 2x 7y 0.4x 2.25y 0.4x 3.1different slopesame slope,same interceptsame slope,different interceptLines intersect.Graphs are same line.Lines are parallel.one solutioninfinitely many solutionsno solutionOften, graphing a system of equations is not the best method of finding itssolution. This is especially true when the solution to the system contains noninteger values. Systems of linear equations can also be solved algebraically. Twocommon ways of solving systems algebraically are the elimination method andthe substitution method. In some cases, one method may be easier to use thanthe other.Example2 Use the elimination method to solve the system of equations.1.5x 2y 202.5x 5y 25One way to solve this system is to multiplyboth sides of the first equation by 5, multiplyboth sides of the second equation by 2, andadd the two equations to eliminate y. Thensolve the resulting equation.5(1.5x 2y) 5(20) 2(2.5x 5y) 2( 25)7.5x 10y 1005x 10y 5012.5x 50x 4Now substitute 4 for x ineither of the originalequations.1.5x 2y 201.5(4) 2y 20 x 42y 14y 7The solution is (4, 7). Check it by substituting into 2.5x 5y 25. If thecoordinates make both equations true, then the solution is correctIf one of the equations contains a variable with a coefficient of 1, the systemcan often be solved more easily by using the substitution method.68Chapter 2Systems of Linear Equations and Inequalities

Example3 Use the substitution method to solve the system of equations.2x 3y 8x y 2You can solve the second equation foreither y or x. If you solve for x, theresult is x y 2. Then substitutey 2 for x in the first equation.2x 3y 82(y 2) 3y 8 x y 25y 44y 4Now substitute for y in either of5the original equations, and solvefor x.x y 24x 2545y 145x 514 45 5The solution is , .GRAPHING CALCULATOR EXPLORATIONYou can use a graphing calculator to find thesolution to an independent system of equations. Graph the equations on the same screen.WHAT DO YOU THINK?3. How accurate are solutions found on thecalculator? Use the CALC menu and select 5:intersect todetermine the coordinates of the point ofintersection of the two graphs.4. What type of system do the equations5x 7y 70 and 10x 14y 120 form?What happens when you try to find theintersection point on the calculator?TRY THESE5. Graph a system of dependent equations.Find the intersection point. Use the TRACEfunction to move the cursor and find theintersection point again. What pattern doyou observe?Find the solution to each system.1. y 500x 202. 3x 4y 320y 20x 5005x 2y 340You can use a system of equations to solve real-world problems. Choose thebest method for solving the system of equations that models the situation.l WoreaAponldRExamplep li c a ti4 CONSUMER CHOICES Refer to the application at the beginning of the lesson.a. What is the break-even point in the two lease plans that Madison isconsidering?b. If Madison keeps the lease for 24 months, which lease should she choose?a. First, write an equation to represent the amount she will pay with each plan.Let C represent the total cost and m the number of months she has had thelease.Lease 1 ( 200 down with monthly payment of 326):C 326m 200Lease 2 ( 1600 down with monthly payment of 226): C 226m 1600Now, solve the system of equations. Since both equations contain C, we cansubstitute the value of C from one equation into the other.(continued on the next page)Lesson 2-1 Solving Systems of Equations in Two Variables69

C 326m 200226m 1600 326m 200 C 226m 16001400 100m14 mWith the fourteenth monthly payment, she reaches the break-even point.b. The graph of the equations shows that after that point, Lease 1 is moreexpensive for the 2-year lease. So, Madison should probably choose Lease 2.C900080007000Cost of 6000Lease 5000(dollars) 4000 C 226m 1600300020001000C 326m 200OC HECKCommunicatingMathematicsFOR246810(14, 4764)12 14Months1618202224 mU N D E R S TA N D I N GRead and study the lesson to answer each question.1. Write a system of equations in which it is easier to use the substitution methodto solve the system rather than the elimination method. Explain your choice.2. Refer to the application at the beginning of the lesson. Explain what factorsMadison might consider before making a decision on which lease to select.3. MathJournal Write a description of the three different possibilities that mayoccur when graphing a system of two linear equations. Include examples andsolutions that occur with each possibility.Guided Practice4. State whether the system 2y 3x 6 and 4y 16 6x is consistent andindependent, consistent and dependent, or inconsistent. Explain your reasoning.Solve each system of equations by graphing.5. y 5x 2y 2x 56. x y 22x 2y 10Solve each system of equations algebraically.7. 7x y 95x y 158. 3x 4y 16x 2y 3139. x y 4325x 4y 1410. Sales HomePride manufactures solid oak racks for displaying baseballequipment and karate belts. They usually sell six times as many baseball racksas karate-belt racks. The net profit is 3 from each baseball rack and 5 fromeach karate-belt rack. If the company wants a total profit of 46,000, how manyof each type of rack should they sell?70Chapter 2 Systems of Linear Equations and Inequalitieswww.amc.glencoe.com/self check quiz

E XERCISESPracticeState whether each system is consistent and independent, consistent anddependent, or inconsistent.A11. x 3y 18 x 2y 712. y 0.5x2y x 413. 35x 40y 557x 8y 11Solve each system of equations by graphing.14. x 515. y 316. x y 217. x 3y 018. y x 219. 3x 2y 64x 5y 20B2x 6y 52x 8x 2y 43x y 10x 12 4y20. Determine what type of solution you would expect from the system of equations3x 8y 10 and 16x 32y 75 without graphing the system. Explain how youdetermined your answer.Solve each system of equations algebraically.21. 5x y 1622. 3x 5y 823. y 6 x24. 2x 3y 325. 3x 10y 526. x 2y 827. 2x 5y 4328. 51 529. 4x 5y 82x 3y 312x 15y 4C3x 6y 5x 2y 12x 7y 24x x 1 65 6y 1x 4.5 y2x y 7y 113x 7y 1030. Find the solution to the system of equations 3x y 9 and 4x 2y 8.31. Explain which method seems most efficient to solve the system of equationsa b 0 and 3a 2b 15. Then solve the system.l WoreaAponldRApplicationsand ProblemSolvingp li c a ti32. Sports Spartan Stadium at San Jose StateUniversity in California has a seatingcapacity of about 30,000. A newspaperarticle states that the Spartans get fourtimes as many tickets as the visiting team.Suppose S represents the number oftickets for the Spartans and V representsthe number of tickets for the visitingteam’s fans.a. Which system could be used by a newspaper reader to determine how manytickets each team gets?A 4S 4V 30,000B S 4V 0S 4VS V 30,000C S V 30,000V 4S 0b. Solve the system to find how many tickets each team gets.33. Geometry Two triangles have the same perimeter of 20 units. One triangle isan isosceles triangle. The other triangle has a side 6 units long. Its other twosides are the same lengths as the base and leg of the isosceles triangle.a. What are the dimensions of each triangle?b. What type of triangle is the second triangle?Lesson 2-1 Solving Systems of Equations in Two Variables71