Download presentation

Presentation is loading. Please wait.

Published byEzra Lynch Modified over 7 years ago

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+ + –

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google