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Solving 3 Variable Linear Equations

Learn how to solve systems of linear equations with 3 variables using the linear combination method. Get step-by-step instructions and examples.

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Solving 3 Variable Linear Equations

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

  2. 3.6 Solving Systems of Linear Equations in 3 Variables10/6/15

  3. 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.

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

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

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

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

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