1. 1.6 Solving Linear Systems in Three Variables 10/23/12

2. Steps Step 1: Eliminate one variable by rewriting the linear system in 3 variables as a linear system in 2 variables using the elimination method. Step 2: Solve the new equations for both of its variables. Step 3: Substitute the value obtained in Step 2 into one of the 3 original equations and solve for the remaining variable. Step 4: Check the solution in each of the original equations.

3. Solve. Example 2 Step 1 Step 2 Step 3

4. Example 2 Use the Linear Combination Method Solve the system. Equation 1 112y2y3x3x = + Equation 2 4y2x2x = – 4z4z+ 3z3z+ 3y3y5x5x – 1 = 5z5z+ – Equation 3

5. Example 2 Use the Linear Combination Method Solve the system. SOLUTION STEP 1Rewrite the system as a system in two variables. First, add 2 times Equation 2 to Equation 1 to eliminate y. Equation 1 112y2y3x3x = + Equation 2 4y2x2x = – 4z4z+ 3z3z+ 3y3y5x5x – 1 = 5z5z+ – Equation 3 == 112y2y3x3x = + 4y2x2x – 4z4z+ 3z3z + 2y2y3x3x = + 82y2y4x4x – 4z4z+ 6z6z + 197x7x = 10z+ New Equation 1

6. Example 2 Use the Linear Combination Method Now add 3 times Equation 2 to Equation 3 to eliminate y. – 4y2x2x = – 3z3z + 3y3y6x6x = + 9z9z 13y3y5x5x = 5z5z+ –– 13y3y5x5x = 5z5z+ –– –– 12 – New Equation 2 x = 4z4z –– 13 – STEP 2Solve the new system of linear equations in two variables. First, add 7 times new Equation 2 to new Equation 1 to eliminate x. == 197x7x = 10z + x 4z4z –– 13 – 197x7x = 10z + 7x7x 28z –– 91 – = 18z –72–

7. Example 2 Use the Linear Combination Method Solve for z. = z 4 Substitute 4 for z in new Equation 1 or 2 and solve for x to get x 3. – = STEP 3Substitute 3 for x and 4 for z in one of the original equations and solve for y. – 4y2x2x = – 3z3z + Equation 2 4y2 = – 3 + Substitute 3 for x and 4 for z. – () 4 () 3 – 4y = – 12 + Multiply. 6 – 4y = – 6+ Combine like terms. 2 = Solve for y. y

8. Example 2 Use the Linear Combination Method STEP 4Check by substituting 3 for x, 2 for y, and 4 for z in each of the original equations. – ANSWER The solution is x 3, y 2, and z 4, or the ordered triple ( 3, 2, 4). – === –

9. Solve the system. Then check your solution. Example 3 3yx = -z- 2y-x = + 5z + 4 yx = 4z4z+ + ANSWER (2, -2, 1)

10. Homework: WS 1.6 #3-6 only