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1.6 Solving Linear Systems in Three Variables

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

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1.6 Solving Linear Systems in Three Variables

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

  4. Example 2 Use the Linear Combination Method Solve the system. Equation 1 3x + 2y + 4z 11 = – Equation 2 2x y 4 + 3z = – – 5x 3y 1 Equation 3 + 5z =

  5. Example 2 SOLUTION STEP 1 Rewrite the system as a system in two variables. First, add 2 times Equation 2 to Equation 1 to eliminate y. 3x + 2y + 4z 11 3x + 2y + 4z 11 = = – – 3z 6z 2x y 4 4x 2y 8 + + = = New Equation1 7x 19 + 10z = Use the Linear Combination Method Solve the system. Equation 1 3x + 2y + 4z 11 = – Equation 2 2x y 4 + 3z = – – 5x 3y 1 Equation 3 + 5z =

  6. Example 2 – – – – 5x 5x 3y 3y + + 5z 5z 1 1 = = – – + – – – – 3z 9z 2x y 4 6x 3y 12 13 13 + = = – – New Equation2 4z x = STEP 2 Solve the new system of linear equations in two variables. First, add 7 times new Equation 2 to new Equation 1 to eliminate x. 10z 10z 7x 19 7x 19 + + = = – – – – – 4z 28z x 7x 91 = = – – 72 18z = Use the Linear Combination Method Now add 3 times Equation 2 to Equation 3 to eliminate y. –

  7. Example 2 Substitute 4 for z in new Equation 1 or 2 and solve for x to get x3. – = STEP 3 Substitute3 for x and 4 for z in one of the original equations and solve for y. – Equation2 – 3z 2x y 4 + = Substitute 3 for x and 4 for z. – 3 2 y 4 + – = – Multiply. – 12 6 y 4 + = ( ) ( ) – 3 4 Combine like terms. – y 4 + 6 = y Solve for y. 2 = Use the Linear Combination Method Solve forz. 4 z =

  8. Example 2 ANSWER The solution is x3, y2, and z4, or the ordered triple ( 3, 2, 4). – = = = – Use the Linear Combination Method STEP 4 Check by substituting 3 for x, 2 for y, and 4 for z in each of the original equations. –

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

  10. Homework: WS 1.6 #3-6 only

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