QuestionWrite the augmented matrix for the given system. The rows of the augmented matrix should be in $\begin{array}{l} 5(x-y)=2-5 y \\ 6 x=2 y+8 \end{array}$
Augmented matrix: $\square$ $\square$ $\square$ $\square$ $\square$ $\square$ $]$ $\square$

Studdy Solution

STEP 1

1. The system of equations needs to be expressed in standard form before writing the augmented matrix.
2. The standard form of a linear equation is $ax + by = c$ .
3. The augmented matrix represents the coefficients of the variables and the constants from the equations.

STEP 2

1. Rearrange each equation into standard form.
2. Write the augmented matrix using the coefficients and constants from the standard form equations.

STEP 3

Rearrange the first equation $5(x-y) = 2 - 5y$ into standard form.
First, distribute the 5 on the left side:
$5x - 5y = 2 - 5y$
Add $5y$ to both sides to move all terms involving $y$ to the left side:
$5x - 5y + 5y = 2$
Simplify:
$5x = 2$
Rearrange to standard form:
$5x + 0y = 2$

STEP 4

Rearrange the second equation $6x = 2y + 8$ into standard form.
Subtract $2y$ from both sides to move all terms involving $y$ to the left side:
$6x - 2y = 8$
This equation is already in standard form.

STEP 5

Write the augmented matrix using the coefficients and constants from the standard form equations.
For the first equation $5x + 0y = 2$ , the coefficients are $5$ and $0$ , and the constant is $2$ .
For the second equation $6x - 2y = 8$ , the coefficients are $6$ and $-2$ , and the constant is $8$ .
The augmented matrix is:
$\begin{bmatrix} 5 & 0 & | & 2 \\ 6 & -2 & | & 8 \end{bmatrix}$

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