4.4 Solve Systems of Equations with Three Variables

Before you get started, take this readiness quiz.

Evaluate $5 x - 2 y + 3 z$ when $x = −2 ,$ $y = −4 ,$ and $z = 3 .$
If you missed this problem, review the reference.

Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. $\begin{cases} - 2 x + y = −11 \\ x + 3 y = 9 \end{cases} .$
If you missed this problem, review the reference.

Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. $\begin{cases} 7 x + 8 y = 4 \\ 3 x - 5 y = 27 \end{cases} .$
If you missed this problem, review the reference.

Determine whether the ordered triple is a solution to the system: $\begin{cases} 3 x + y + z = 2 \\ x + 2 y + z = −3 \\ 3 x + y + 2 z = 4 \end{cases} .$

ⓐ $\left(1 , −3 , 2\right)$ ⓑ $\left(4 , −1 , −5\right)$

Determine whether the ordered triple is a solution to the system: $\begin{cases} x - 3 y + z = −5 \\ - 3 x - y - z = 1 \\ 2 x - 2 y + 3 z = 1 \end{cases} .$

ⓐ $\left(2 , −2 , 3\right)$ ⓑ $\left(−2 , 2 , 3\right)$

Solve the system by elimination: $\begin{cases} 3 x + y - z = 2 \\ 2 x - 3 y - 2 z = 1 \\ 4 x - y - 3 z = 0 \end{cases} .$

Solve the system by elimination: $\begin{cases} 4 x + y + z = −1 \\ - 2 x - 2 y + z = 2 \\ 2 x + 3 y - z = 1 \end{cases} .$

Solve: $\begin{cases} 3 x - 4 z = −1 \\ 2 y + 3 z = 2 \\ 2 x + 3 y = 6 \end{cases} .$

Solve: $\begin{cases} 4 x - 3 z = −5 \\ 3 y + 2 z = 7 \\ 3 x + 4 y = 6 \end{cases} .$

Solve the system of equations: $\begin{cases} x + 2 y + 6 z = 5 \\ - x + y - 2 z = 3 \\ x - 4 y - 2 z = 1 \end{cases} .$

Solve the system of equations: $\begin{cases} 2 x - 2 y + 3 z = 6 \\ 4 x - 3 y + 2 z = 0 \\ - 2 x + 3 y - 7 z = 1 \end{cases} .$

Solve the system by equations: $\begin{cases} x + y - z = 0 \\ 2 x + 4 y - 2 z = 6 \\ 3 x + 6 y - 3 z = 9 \end{cases} .$

Solve the system by equations: $\begin{cases} x - y - z = 1 \\ - x + 2 y - 3 z = −4 \\ 3 x - 2 y - 7 z = 0 \end{cases} .$

The community college fine arts department sold three kinds of tickets to its latest dance presentation. The adult tickets sold for $20, the student tickets for $12 and the child tickets for $10.The fine arts department was thrilled to have sold 350 tickets and brought in $4,650 in one night. The number of child tickets sold is the same as the number of adult tickets sold. How many of each type di…

The community college soccer team sold three kinds of tickets to its latest game. The adult tickets sold for $10, the student tickets for $8 and the child tickets for $5. The soccer team was thrilled to have sold 600 tickets and brought in $4,900 for one game. The number of adult tickets is twice the number of child tickets. How many of each type did the soccer team sell?

Determine whether the ordered triple is a solution to the system.

$\begin{cases} - 3 x + y + z = −4 \\ - x + 2 y - 2 z = 1 \\ 2 x - y - z = −1 \end{cases}$

ⓐ $\left(−5 , −7 , 4\right)$ ⓑ $\left(5 , 7 , 4\right)$

Determine whether the ordered triple is a solution to the system.

$\begin{cases} x + 3 y - z = 15 \\ y = \frac{2}{3} x - 2 \\ x - 3 y + z = −2 \end{cases}$

ⓐ $\left(−6 , 5 , \frac{1}{2}\right)$ ⓑ $\left(5 , \frac{4}{3} , −3\right)$

Solve the system of equations.

$\begin{cases} 6 x - 5 y + 2 z = 3 \\ 2 x + y - 4 z = 5 \\ 3 x - 3 y + z = −1 \end{cases}$

Solve the system of equations.

$\begin{cases} 5 x - 3 y + 2 z = −5 \\ 2 x - y - z = 4 \\ 3 x - 2 y + 2 z = −7 \end{cases}$

Solve the system of equations.

$\begin{cases} 4 x - 3 y + z = 7 \\ 2 x - 5 y - 4 z = 3 \\ 3 x - 2 y - 2 z = −7 \end{cases}$

Solve the system of equations.

$\begin{cases} 11 x + 9 y + 2 z = −9 \\ 7 x + 5 y + 3 z = −7 \\ 4 x + 3 y + z = −3 \end{cases}$

Solve the system of equations.

$\begin{cases} x + \frac{1}{2} y + \frac{1}{2} z = 0 \\ \frac{1}{5} x - \frac{1}{5} y + z = 0 \\ \frac{1}{3} x - \frac{1}{3} y + 2 z = −1 \end{cases}$

Solve the system of equations.

$\begin{cases} \frac{1}{3} x - y + \frac{1}{2} z = 4 \\ \frac{2}{3} x + \frac{5}{2} y - 4 z = 0 \\ x - \frac{1}{2} y + \frac{3}{2} z = 2 \end{cases}$

Solve the system of equations.

$\begin{cases} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{cases}$

Solve the system of equations.

$\begin{cases} 3 x - z = −3 \\ 5 y + 2 z = −6 \\ 4 x + 3 y = −8 \end{cases}$

Solve the system of equations.

$\begin{cases} x - 2 y + 2 z = 1 \\ - 2 x + y - z = 2 \\ x - y + z = 5 \end{cases}$

Solve the system of equations.

$\begin{cases} x + 4 y + z = −8 \\ 4 x - y + 3 z = 9 \\ 2 x + 7 y + z = 0 \end{cases}$

Solve the system of equations.

$\begin{cases} x + y - 2 z = 3 \\ - 2 x - 3 y + z = −7 \\ x + 2 y + z = 4 \end{cases}$

Solve the system of equations.

$\begin{cases} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{cases}$

Solve the given problem.

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.

Solve the given problem.

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of $65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of $140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of $250.…

How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?