Presentation on theme: "Module 6-3 Objectives Solve systems of linear equations in two variables by elimination. Compare and choose an appropriate method for solving systems of."— Presentation transcript:
1 Module 6-3ObjectivesSolve systems of linear equations in two variables by elimination.Compare and choose an appropriate method for solving systems of linear equations.
2 Another method for solving systems of equations is elimination Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable.When you use the elimination method to solve a system of linear equations, align all like terms in the equations. Then determine whether any like terms can be eliminated because they have opposite coefficients.
3 Solving Systems of Equations by Elimination Step 1Write the system so that like terms are aligned.Step 2Eliminate one of the variables and solve for the other variable.Step 3Substitute the value of the variable into one of the original equations and solve for the other variable.Write the answers from Steps 2 and 3 as an ordered pair, (x, y), and check.Step 4
4 Later in this lesson you will learn how to multiply one or more equations by a number in order to produce opposites that can be eliminated.
5 Example 1A: Elimination Using Addition 3x – 4y = 10Solve by elimination.x + 4y = –2Step 13x – 4y = 10Align like terms. −4y and +4y are opposites.x + 4y = –2Step 2Add the equations to eliminate y.4x = 84x = 8Simplify and solve for x.4x = 8x = 2Divide both sides by 4.
6 Example 1A ContinuedStep 3x + 4y = –2Write one of the original equations.2 + 4y = –2Substitute 2 for x.– –24y = –4Subtract 2 from both sides.4y –4y = –1Divide both sides by 4.Step 4 (2, –1)Write the solution as an ordered pair.
7 Check It Out! Example 1By + 3x = –2Solve by elimination.2y – 3x = 14Align like terms. 3x and −3x are opposites.Step 12y – 3x = 14y + 3x = –2Add the equations to eliminate x.Step 23y = 123y = 12Simplify and solve for y.y = 4Divide both sides by 3.
8 Check It Out! Example 1B Continued Write one of the original equations.Step 3 y + 3x = –24 + 3x = –2Substitute 4 for y.– –43x = –6Subtract 4 from both sides.Divide both sides by 3.3x = –6x = –2Write the solution as an ordered pair.Step 4(–2, 4)
9 Remember to check by substituting your answer into both original equations.
10 In some cases, you will first need to multiply one or both of the equations by a number so that one variable has opposite coefficients.
11 Check It Out! Example 23x + 3y = 15Solve by elimination.–2x + 3y = –53x + 3y = 15–(–2x + 3y = –5)Step 1Both equations contain 3y. Add the opposite of each term in the second equation.3x + 3y = 15+ 2x – 3y = +5Step 25x = 20Eliminate y.5x = 20x = 4Simplify and solve for x.
12 Check It Out! Example 2 Continued Write one of the original equations.Step 33x + 3y = 153(4) + 3y = 15Substitute 4 for x.12 + 3y = 15– –123y = 3Subtract 12 from both sides.Simplify and solve for y.y = 1Write the solution as an ordered pair.(4, 1)Step 4
13 Example 3A: Elimination Using Multiplication First Solve the system by elimination.x + 2y = 11–3x + y = –5Multiply each term in the second equation by –2 to get opposite y-coefficients.x + 2y = 11Step 1–2(–3x + y = –5)x + 2y = 11+(6x –2y = +10)Add the new equation to the first equation to eliminate y.7x = 21Step 27x = 21x = 3Solve for x.
14 Example 3A ContinuedWrite one of the original equations.Step 3x + 2y = 113 + 2y = 11Substitute 3 for x.– –32y = 8Subtract 3 from both sides.Solve for y.y = 4Step 4(3, 4)Write the solution as an ordered pair.
15 Example 3B: Elimination Using Multiplication First Solve the system by elimination.–5x + 2y = 322x + 3y = 10Multiply the first equation by 2 and the second equation by 5 to get opposite x-coefficientsStep 12(–5x + 2y = 32)5(2x + 3y = 10)–10x + 4y = 64+(10x + 15y = 50)Add the new equations toeliminate x.Step 219y = 114y = 6Solve for y.
16 Example 3B ContinuedWrite one of the original equations.Step 32x + 3y = 102x + 3(6) = 10Substitute 6 for y.2x + 18 = 10–18 –182x = –8Subtract 18 from both sides.x = –4Solve for x.Step 4Write the solution as an ordered pair.(–4, 6)
17 Example 4: ApplicationPaige has $7.75 to buy 12 sheets of felt and card stock for her scrapbook. The felt costs $0.50 per sheet, and the card stock costs $0.75 per sheet. How many sheets of each can Paige buy?Write a system. Use f for the number of felt sheets and c for the number of card stock sheets.0.50f c = 7.75The cost of felt and card stock totals $7.75.f + c = 12The total number of sheets is 12.
18 Example 4 ContinuedStep 10.50f c = 7.75Multiply the second equation by –0.50 to get opposite f-coefficients.+ (–0.50)(f + c) = 120.50f c = 7.75Add this equation to the first equation to eliminate f.+ (–0.50f – 0.50c = –6)0.25c = 1.75Step 2c = 7Solve for c.Write one of the original equations.Step 3f + c = 12f + 7 = 12Substitute 7 for c.–7 –7f = 5Subtract 7 from both sides.
19 Example 4 ContinuedStep 4(7, 5)Write the solution as an ordered pair.Paige can buy 7 sheets of card stock and 5 sheets of felt.
20 All systems can be solved in more than one way All systems can be solved in more than one way. For some systems, some methods may be better than others.