Advanced Algebra / Trigonometry

Section 3-5Systems of Equations in Three

Variables

Target Goals

1) Solve systems of linear equations in three variables.

Systems of Equations in Three Variables

:Ex5 3 2 2x y z 2 5x y z

4 2 16x y z

A solution to a system of equations in three variables is the ordered triple , , that satisfies all three equations.x y z

We can solve a system of equations in three variables using an extensionof the elimination method...

1) Pick two equations and "eliminate" one of the variables.2) Pick a different two equations and "eliminate" the same variable.

Now you have two equations with two variables.3) Solve this new system of two equations for both variables using prior

techniques.4) Substitute the values found into one of the original equations to solve

for the third variable.

Solve the system of equations.

Multiply the 2nd equation by 2.

Example 1 5 3 2 2x y z 2 5x y z

4 2 16x y z Eliminate in the 1st and 2nd equations.z

2 5 2x y z

5 3 2 2x y z 4 2 2 10x y z 9 5 12x y

Eliminate in the 2nd and 3rd equations.zMultiply the 2nd equation by 2.

4 2 16x y z 4 2 2 10x y z

5 6 26x y System of two equations

9 5 12x y 5 6 26x y

Eliminate by multiplying 1st equation by 6 and 2nd equation by 5.y

12 69 5x y 55 6 26x y

54 30 72x y 25 30 130x y

29 58x 2x

Substitute back in to solve for .x y

54 32 0 72y 108 30 72y

30 180y 6y

Substitute and back in to solve for .x y z 2 65 3 2 2z

10 18 2 2z 8 2 2z

2 6z 3z

Solution 2,6, 3

Solve the system of equations.

Multiply the 1st equation by 2

Example 2 4 3 6 18x y z 5 4 48x y z

6 2 5 0x y z Eliminate in the 1st and 2nd equations.z

4 3 6 18 2x y z

8 6 12 36x y z 3 15 12 144x y z 11 21 108x y

Eliminate in the 2nd and 3rd equations.zMultiply the 2nd equation by 5

24 8 20 0x y z 5 25 20 240x y z

29 33 240x y System of two equations

11 21 108x y 29 33 240x y

Eliminate by multiplying 1st equation by 11 and 2nd equation by 7.y

11 21 08 11 1x y 29 33 7240x y

121 231 1188x y 203 231 1680x y

82 492x 6x

Substitute back in to solve for .x y

121 231 86 11 8y 726 231 1188y

231 462y 2y

Substitute and back in to solve for .x y z 4 36 6 12 8z

24 6 6 18z 30 6 18z

6 48z 8z Solution 6,2,8

and the 2nd equation by 3.

5 4 4 38x y z

and the 3rd equation by 4.

5 4 4 58x y z 6 2 5 0 4x y z

Exit Slip

.A

.B

Target Goal: Solve systems of linear equations in three variables.

.C

.D

2 4 5 18x y z What is the value of in the solution to the system of equations.y

3 5 2 27x y z 5 3 17x y z

1y

3y

2y

4y

End

Advanced Algebra / Trigonometry

Section 3-5Systems of Equations in Three

Variables

Target Goals

1) Solve systems of linear equations in three variables.

Systems of Equations in Three Variables

:Ex5 3 2 2x y z 2 5x y z

4 2 16x y z

A solution to a system of equations in three variables is the ordered triple , , that satisfies all three equations.x y z

We can solve a system of equations in three variables using an extensionof the elimination method...

1) Pick two equations and "eliminate" one of the variables.2) Pick a different two equations and "eliminate" the same variable.

Now you have two equations with two variables.3) Solve this new system of two equations for both variables using prior

techniques.4) Substitute the values found into one of the original equations to solve

for the third variable.

Solve the system of equations.Example 1 5 3 2 2x y z

2 5x y z 4 2 16x y z

Solve the system of equations.Example 2 4 3 6 18x y z

5 4 48x y z 6 2 5 0x y z

Exit Slip

.A

.B

Target Goal: Solve systems of linear equations in three variables.

.C

.D

2 4 5 18x y z What is the value of in the solution to the system of equations.y

3 5 2 27x y z 5 3 17x y z

1y

3y

2y

4y