4.5 Solve Systems of Equations Using Matrices

Augmented matrix notation, the three legal row operations, and row reduction to echelon form as a bookkeeping-light route to the solution.

Intermediate Algebra 2e, 4.5 Solve Systems of Equations Using Matrices

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37 exercises

4.5 Exercise 1

Before you get started, take this readiness quiz.

Solve: $3 \left(x + 2\right) + 4 = 4 \left(2 x - 1\right) + 9 .$
If you missed this problem, review the reference.

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4.5 Exercise 2

Solve: $0.25 p + 0.25 \left(p + 4\right) = 5.20 .$
If you missed this problem, review the reference.

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4.5 Exercise 3

Evaluate when $x = −2$ and $y = 3 : 2 x^{2} - x y + 3 y^{2} .$
If you missed this problem, review the reference.

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4.5 Exercise 5

Write each system of linear equations as an augmented matrix:

ⓐ $\begin{cases} 3 x + 8 y = −3 \\ 2 x = −5 y - 3 \end{cases}$ ⓑ $\begin{cases} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = −3 \end{cases}$

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4.5 Exercise 6

Write each system of linear equations as an augmented matrix:

ⓐ $\begin{cases} 11 x = −9 y - 5 \\ 7 x + 5 y = −1 \end{cases}$ ⓑ $\begin{cases} 5 x - 3 y + 2 z = −5 \\ 2 x - y - z = 4 \\ 3 x - 2 y + 2 z = −7 \end{cases}$

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4.5 Exercise 8

Write the system of equations that corresponds to the augmented matrix: $\begin{bmatrix} 1 & & & −1 & & & 2 & & & 3 \\ 2 & & & 1 & & & −2 & & & 1 \\ 4 & & & −1 & & & 2 & & & 0 \end{bmatrix} .$

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4.5 Exercise 9

Write the system of equations that corresponds to the augmented matrix: $\begin{bmatrix} 1 & & & 1 & & & 1 & & & 4 \\ 2 & & & 3 & & & −1 & & & 8 \\ 1 & & & 1 & & & −1 & & & 3 \end{bmatrix} .$

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4.5 Exercise 11

Perform the indicated operations sequentially on the augmented matrix:

ⓐ Interchange rows 1 and 3.

ⓑ Multiply row 3 by 3.

$\left[\right. \begin{matrix}5 & & & −2 & & & −2 & & \\ 4 & & & −1 & & & −4 & & \\ −2 & & & 3 & & & 0 & & \end{matrix} \left|\right. \begin{matrix} & & −2 \\ & & 4 \\ & & −1\end{matrix} \left]\right.$

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4.5 Exercise 12

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 1 and 2,

ⓑ Multiply row 1 by 2,

$\left[\right. \begin{aligned}2 & & & −3 & & & −2 & & \\ 4 & & & 1 & & & −3 & & \\ 5 & & & 0 & & & 4 & & \end{aligned} \left|\right. \begin{matrix} & & −4 \\ & & 2 \\ & & −1\end{matrix} \left]\right.$

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4.5 Exercise 14

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: $\left[\right. \begin{aligned}1 & & & −1 & & \\ 3 & & & −6 & & \end{aligned} \left|\right. \begin{matrix} & & 2 \\ & & 2\end{matrix} \left]\right. .$

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4.5 Exercise 15

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: $\left[\right. \begin{matrix}1 & & & −1 & & \\ −2 & & & −3 & & \end{matrix} \left|\right. \begin{matrix} & & 3 \\ & & 2\end{matrix} \left]\right. .$

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4.5 Exercise 17

Solve the system of equations using a matrix: $\begin{cases} 2 x + y = 7 \\ x - 2 y = 6 \end{cases} .$

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4.5 Exercise 18

Solve the system of equations using a matrix: $\begin{cases} 2 x + y = −4 \\ x - y = −2 \end{cases} .$

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4.5 Exercise 20

Solve the system of equations using a matrix: $\begin{cases} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = −3 \end{cases} .$

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4.5 Exercise 21

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

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4.5 Exercise 23

Solve the system of equations using a matrix: $\begin{cases} x - 2 y + 2 z = 1 \\ −2 x + y - z = 2 \\ x - y + z = 5 \end{cases} .$

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4.5 Exercise 24

Solve the system of equations using a matrix: $\begin{cases} 3 x + 4 y - 3 z = −2 \\ 2 x + 3 y - z = −12 \\ x + y - 2 z = 6 \end{cases} .$

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4.5 Exercise 26

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

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4.5 Exercise 27

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

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4.5 Exercise 29

Write each system of linear equations as an augmented matrix.

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

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4.5 Exercise 31

Write each system of linear equations as an augmented matrix.

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

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4.5 Exercise 33

Write the system of equations that corresponds to the augmented matrix.

$\left[\right. \begin{aligned}2 & & & −4 \\ 3 & & & −3\end{aligned} \left|\right. \begin{matrix}−2 \\ −1\end{matrix} \left]\right.$

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4.5 Exercise 35

Write the system of equations that corresponds to the augmented matrix.

$\left[\right. \begin{aligned}2 & & & −2 & & & 0 \\ 0 & & & 2 & & & −1 \\ 3 & & & 0 & & & −1\end{aligned} \left|\right. \begin{matrix}−1 \\ 2 \\ −2\end{matrix} \left]\right.$

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4.5 Exercise 37

Perform the indicated operations on the augmented matrices.

$\left[\right. \begin{aligned}4 & & & −6 \\ 3 & & & 2\end{aligned} \left|\right. \begin{matrix}−3 \\ 1\end{matrix} \left]\right.$

ⓐ Interchange rows 1 and 2

ⓑ Multiply row 1 by 4

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4.5 Exercise 39

Perform the indicated operations on the augmented matrices.

$\left[\right. \begin{aligned}6 & & & −5 & & & 2 \\ 2 & & & 1 & & & −4 \\ 3 & & & −3 & & & 1\end{aligned} \left|\right. \begin{matrix}3 \\ 5 \\ −1\end{matrix} \left]\right.$

ⓐ Interchange rows 2 and 3

ⓑ Multiply row 2 by 5

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4.5 Exercise 41

Perform the indicated operations on the augmented matrices.

Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: $\left[\right. \begin{aligned}1 & & & −2 & & & 3 \\ 3 & & & −1 & & & −2 \\ 2 & & & −3 & & & −4\end{aligned} \left|\right. \begin{matrix}−4 \\ 5 \\ −1\end{matrix} \left]\right. .$