1. Goal: Solve systems of linear equations in three variables

4. Example 1 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

5. Example 1 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–

6. Example 1 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

7. Example 1 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). – === –

11. f. (1,1,-3)g. (-2, 4, 0) h. (0, -3, 10) j. (-3, 0, -2)

16. Example 2 Solve a System with No Solution Solve the system. Equation 1 2yx = + Equation 2 143y3y3x3x = + z+ 3z3z+ 2y2yx – 4 = z+ Equation 3 SOLUTION Multiply Equation 1 by 3 and add the result to Equation 2. – 143y3y3x3x = + 3z3z+ to Equation 2. Add 3 times Equation 1 63y3y3x3x = 3z3z –––– – 8 = 0 False statement

17. Example 2 Solve a System with No Solution ANSWER Because solving the system resulted in the false statement 0 8, the original system of equations has no solution. =

18. Example 3 Solve a System with Infinitely Many Solutions Solve the system. Equation 1 4yx = + Equation 2 z+ Equation 3 4yx = +z – 123y3y3x3x = + z+ SOLUTION STEP 1Rewrite the system as a system in two variables. Add Equation 1 4yx = +z+ to Equation 2. 4yx = +z – New Equation 1 82y2y2x2x = +

19. Example 3 Solve a System with Infinitely Many Solutions Add Equation 2 New Equation 2 164y4y4x4x = + 4yx = +z – to Equation 3. 123y3y3x3x = + z+ STEP 2Solve the new system of linear equations in two variables. Add 2 times new Equation 1 164y4y4x4x = ––– – to new Equation 2. 164y4y4x4x = + 00 =

20. Example 3 Solve a System with Infinitely Many Solutions ANSWER Because solving the system resulted in the true statement 0 0, the original system of equations has infinitely many solutions. The three planes intersect in a line. =

21. Checkpoint Tell how many solutions the linear system has. If the system has one solution, solve the system. Then check your solution. Solve Systems 1yx = +3z3z+ 1yx = +z – 13y3yx = 4z4z+ ––– 1. 52y2y2x2x = 2z2z+ – 2. 4 y x = z+ – 2y3x3x = +2z2z –– ANSWER 1 ; (1, 0, 0) ANSWER no solution

22. x + y + 3z = 1 x + y – z = 1 -x – 3y + 4z = -1

23. x – y + z = 4 3x + y – 2z = -2 2x – 2y + 2z = 5

24. x + y + 2z = 10 -x + 2y + z = 5 -x + 4y + 3z = 15

25. Checkpoint Tell how many solutions the linear system has. If the system has one solution, solve the system. Then check your solution. Solve Systems 10yx = +2z2z+ 3. ANSWER infinitely many solutions 52y2yx = z+ + – 154y4yx = 3z3z+ + –