1. 3.5 Solving Linear Systems in Three Variables 10/4/13

2. Intersection of 3 planes We’ve been solving system of equations in 2 variables. The solution is a point where the lines intersect. For systems of equations with 3 variables, the solution is a point where all 3 planes intersect.

3. Solve: Notice Eqn1 has only 2 variables. Solve for one variable (x). Substitute -3z +5 for x in the other 2 equations. Example 1

4. Dist. Prop Combine Like terms New Eqn 2 New Eqn 3

5. New Eqn 2 New Eqn 3 Solve by Elimination 7( ) -2( ) + + Solution (x, y, z) (-4, -1, 3) Substitute z= 3 in any of the new Eqns.

6. Check the Solution (-4, -1, 3)

7. Solve the system: Step 1: Pick any 2 original equations and eliminate a variable. Eliminate the same variable from a second pair of original equations. Step 2: With the 2 new equations from Step 1 eliminate one of the 2 variables and solve for the remaining variable. Substitute the value you obtained for the variable into one of the 2 new equations and solve for the other variable. Step 3: Substitute the values of the 2 variables obtained in Step 2 into one of the 3 original equations and solve for the last variable (the one you eliminated in step 1). Step 4: Check the solution in each of the original equations.

8. Solve. Example 2 Step 1 Step 2 Step 3 -1( ) New Eqn 1 New Eqn 2

9. Example 3 Solve the system. Equation 1 112y2y3x3x = + Equation 2 4y2x2x = – 4z4z+ 3z3z+ 3y3y5x5x – 1 = 5z5z+ – Equation 3 ( 3, 2, 4). ANSWER

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

11. Homework: 3.5 p.156 #7, 16-19