3.5 Solving Linear Systems in Three Variables

10/4/13

Intersection of 3 planesWe’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:

Equation 1Equation 2

Equation 3

Notice Eqn1 has only 2 variables. Solve for one variable (x).

Substitute -3z +5 for x in the other 2 equations.

Example 1

Dist. Prop

Combine Like terms

New Eqn 2New Eqn 3

New Eqn 2New Eqn 3

Solve by Elimination

7( )-2( )

+ +−48 𝑧=−144−48=−48𝑧=3

Substitute z = 3 in

Solution (x, y, z)(-4, -1, 3)

Substitute z= 3 in any of the new Eqns.

Check the Solution (-4, -1, 3)

2x y z 33x 2y z 2 x 3y 2z 1

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.

2x y z 33x 2y z 2 x 3y 2z 1

2x y z 33x 2y z 2

5 3 5 x y3x 2y z 2 x 3y 2z 1

2                      6x 4y 2z 4 x 3y 2z 15 7 5 x y5 3 5

5 7 5 x yx y

4 0 y 5 507 x

1x

02 1( ) 3 z

Solve.Example 2

Step 1

Step 1

Step 2

Step 3-1( )

New Eqn 1

New Eqn 2

𝑦=0 𝑧=1𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 :(1 ,0 ,1)

Example 3

Solve the system.Equation 1112y3x =+

Equation 24y2x =–4z+

3z+3y5x – 1=5z+ – Equation 3

( 3, 2, 4).ANSWER

Solve the system. Then check your solution.

Example 4

3yx =- z--12y-x =+ 5z+4yx =4z++

ANSWER (2, -2, 1)

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