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Q57.

Expert-verified

Algebra 1

Found in: Page 392

Book edition Student Edition

Author(s) Carter, Cuevas, Day, Holiday, Luchin

Pages 801 pages

ISBN 9780078884801

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Short Answer

Use an augmented matrix to solve each system of equations.
$\begin{array}{l} x - y = 5 \\ 3 x + 3 y = 3 \end{array}$

The solution of the system of equations is $(3, - 2)$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Write the augmented matrix.

To write the equations in augmented matrix place the coefficients of the equations and the constant terms into a matrix separated by a dashed line.

Here, the augmented matrices are:

$[\begin{matrix} 1 & - 1 & 5 \\ 3 & 3 & 3 \end{matrix}]$

Step 2. Use row operations to solve the system.

To make the first element in the second row a 0, divide the second row by 3 and then subtract row 1 from the resultant row 2

$\begin{array}{r} [\begin{matrix} 1 & - 1 & 5 \\ 3 & 3 & 3 \end{matrix}] \overset{\frac{1}{3} R_{2}}{\to} [\begin{matrix} 1 & - 1 & 5 \\ 1 & 1 & 1 \end{matrix}] \\ \overset{R_{2} - R_{1}}{\to} [\begin{matrix} 1 & - 1 & 5 \\ 0 & 2 & - 4 \end{matrix}] \end{array}$

To make the second element in the second row a 1, divide the second row by 2.

$[\begin{matrix} 1 & - 1 & 5 \\ 0 & 2 & - 4 \end{matrix}] \overset{\frac{1}{2} R_{2}}{\to} [\begin{matrix} 1 & - 1 & 5 \\ 0 & 1 & - 2 \end{matrix}]$

Step 3. Row reduce the matrix.

Further row-reduce the augmented matrix, by making the second element in the first row a zero.

$[\begin{matrix} 1 & - 1 & 5 \\ 0 & 1 & - 2 \end{matrix}] \overset{R_{1} + R_{2}}{\to} [\begin{matrix} 1 & 0 & 3 \\ 0 & 1 & - 2 \end{matrix}]$ $[\begin{matrix} 1 & - 1 & 5 \\ 0 & 1 & - 2 \end{matrix}] \overset{R_{1} + R_{2}}{\to} [\begin{matrix} 1 & 0 & 3 \\ 0 & 1 & - 2 \end{matrix}]$

Here, the first row will give the solution of $x$ , because the coefficient of $y$ is 0 and the coefficient of $x$ is 1. Similarly, the second row will give the solution of $y$ , because the coefficient of $x$ is 0 and the coefficient of $y$ is 1. Therefore, the solution is $(3, - 2)$ .

Most popular questions for Math Textbooks

Shelly has $$ 175$ to shop for jeans and sweaters. Each pair of jeans cost $$ 25$ , each sweater costs $$ 20$ , and she buys 8 items. Determine the number of pairs of jeans and sweaters Shelly bought.

Solve each equation or formula for the variable specified.

$2 x + 4 y = 12$ , for x

Use substitution to solve each system of equations.

$\begin{array}{l} x + 3 y = 9 \\ x + y = 1 \end{array}$

Use elimination to solve each system of equations.

$\begin{array}{l} - 7 x + 3 y = 12 \\ 2 x - 8 y = - 32 \end{array}$

Use substitution to solve each system of equations.

$\begin{array}{l} x - y = 8 \\ y = - 3 x \end{array}$

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