3.3 Solving Linear Systems by Linear Combination 10/12/12

Solving Systems of Equations So far, we have solved systems using graphing and substitution. The third method is called Linear Combination or Elimination.

Linear Combination or Elimination Is a method of solving a system of equations by multiplying one or both equations by a constant, then adding the revised equations to eliminate a variable

Linear Combination Method Step 1: Multiply one or both of the equations by a constant, if necessary, to obtain coefficients that differ only in sign for one of its variables. Step 2: Add the revised equations from Step 1. Combining like terms will eliminate one variable. Solve for the remaining variable. Step 3: Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable. Step 4: Check the solution in each of the original equations.

Solve Solution (2, -1) Method: by Graphing

Solve Solution (2, -1) Method: Substitution

Solve Solution (2, -1) Method: Linear Combination Step 1: Multiply one or both of the equations by a constant, if necessary, to obtain coefficients that differ only in sign for one of its variables. + Step 2: Add the revised equations from Step 1. Combining like terms will eliminate one variable. Solve for the remaining variable. 5x = 10 5 x = y = y = 2 -2 y = -1 Step 3: Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

Example 1 Multiply One Equation Solve the linear system using the linear combination method. Equation 1 63y3y2x2x = – Equation 2 85y5y4x4x = – SOLUTION STEP 1Multiply the first equation by 2 so that the coefficients of x differ only in sign. – 63y3y2x2x = – 85y5y4x4x = –85y5y4x4x = – 126y6y4x4x = +–– 4y = – STEP 2Add the revised equations and solve for y.

Example 1 Multiply One Equation STEP 3Substitute 4 for y in one of the original equations and solve for x. – Write Equation 1. 63y3y2x2x = – Substitute 4 for y. 62x2x = – () 4 – 3 – Simplify. 6122x2x = + Subtract 12 from each side. 62x2x = – Solve for x. 3x = – STEP 4Check by substituting 3 for x and 4 for y in the original equations. –– ANSWER The solution is. () 3,3, –– 4

Example 2 Multiply Both Equations Solve the system using the linear combination method. Equation 1 Equation 2 SOLUTION STEP 1Multiply the first equation by 2 and the second equation by y7x7x = –– 148y8y5x5x = +– 2212y7x7x = –– 148y8y5x5x = + – 4424y14x = –– 4224y15x = + – 2x = –– STEP 2Add the revised equations and solve for x. 2x =

Example 2 Multiply Both Equations STEP 3Substitute 2 for x in one of the original equations and solve for y. 148y8y5x5x = + – Write Equation y8y = + Substitute 2 for x. – () y8y10 = + – Multiply. 3y = Solve for y. STEP 4Check by substituting 2 for x and 3 for y in the original equations. ANSWER The solution is (2, 3).

Checkpoint 1. Solve the system using the linear combination method. Solve a Linear System 54y4yx = – 1y2x2x = + ANSWER (1, 1) – 2. 22y2y3x3x = – 13y3y4x4x = – ANSWER (4, 5 )

Homework: 3.3 p.142 #8-18