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3.5 Solving Linear Systems in Three Variables

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

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3.5 Solving Linear Systems in Three Variables

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

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

  5. 7( ) -2( ) New Eqn 2 New Eqn 3 Solve by Elimination Substitute z = 3 in + + 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. Example 2 Solve. Step 1 New Eqn 1 Step 1 Step 2 New Eqn 2 Step 3 -1( )

  9. Example 3 Solve the system. ANSWER Equation 1 3x + 2y + 4z 11 = ( 3, 2, 4). – Equation 2 2x y 4 + 3z = – – 5x 3y 1 Equation 3 + 5z =

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

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

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