WarmUp Solve using the linear combination method 7

Warm-Up Solve using the linear combination method. 7 x + y =10 3 x -2 y = -3 (1, 3)

3. 6 Solving Systems of Linear Equations in 3 Variables 10/6/15

A system of linear equations in 3 variables Looks something like this: A solution is an ordered triple (x, y, z) that makes all 3 equations true.

Here is a system of three linear equations in three variables: Is the ordered triple (2, -1, 1) a solution?

Steps for solving in 3 variables 1. Use the linear combination method to rewrite the linear system in 3 variables as a linear system in 2 variables. 2. Solve the new linear system for both of its variables. 3. Substitute the values found in Step 2 into one of the original equations and solve for the remaining variable.

Solve the system 1. x + 3 y − z = − 11 2 x + y + z = 1 z’s are easy to cancel! 3 x +4 y = − 10 2. 2 x + y + z =1 5 x − 2 y +3 z = 21 Must cancel z’s again! − 6 x − 3 y − 3 z = − 3 5 x − 2 y +3 z = 21 −x − 5 y = 18 x + 3 y − z = − 11 2 x + y + z = 1 5 x − 2 y + 3 z = 21 3. 3 x +4 y = − 10 −x − 5 y = 18 Solve for x & y 3 x + 4 y = − 10 − 3 x − 15 y = 54 3 x +4(− 4)= − 10 x=2 2(2) +(− 4) +z =1 4 − 4+x =1 z=1 − 11 y = 44 y = − 4 (2, − 4, 1)

Solve the system −x +2 y +z = 3 2 x + 2 y +z = 5 4 x +4 y +2 z = 6 1. −x + 2 y +z = 3 2 x + 2 y + z = 5 z’s are easy to cancel − x + 2 y +z = 3 − 2 x − 2 y −z = − 5 − 3 x = − 2 x = 2/3 2. 2 x +2 y + z = 5 4 x +4 y +2 z= 6 Cancel the z’s again − 4 x − 4 y − 2 z = − 10 4 x +4 y +2 z = 6 0 = − 4 Doesn’t make sense No solution