Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Substitute teachers with Newport School District get paid by the day, although subs with teaching credentials earn a different amount than subs without credentials. Yesterday, $1$ credentialed sub taught in the district. That cost the district $\$89$ . Today, $9$ non-credentialed subs and $11$ credentialed subs taught, receiving $\$1,528$ from the district. How much do subs get paid? $\newline$ Subs without credentials get paid $\_\_\_\_$ per day, and subs with credentials get paid $\_\_\_\_$ per day.

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Grade 8

Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

\newline

Substitute teachers with Newport School District get paid by the day, although subs with teaching credentials earn a different amount than subs without credentials. Yesterday,

1

credentialed sub taught in the district. That cost the district

\$89

. Today,

9

non-credentialed subs and

11

credentialed subs taught, receiving

\$1,528

from the district. How much do subs get paid?

\newline

Subs without credentials get paid

\_\_\_\_

per day, and subs with credentials get paid

\_\_\_\_

per day.

Define Variables: Let $x$ be the daily pay for subs without credentials and $y$ be the daily pay for subs with credentials. From the first day, we have the equation $y = 89$ . From the second day, we have the equation $9x + 11y = 1528$ .
Write System of Equations: Now we write the system of equations: $\newline$ $1$ . $y = 89$ $\newline$ $2$ . $9x + 11y = 1528$
Create Augmented Matrix: To solve using an augmented matrix, we rewrite the system as: $\newline$ $1$ . $0x + 1y = 89$ $\newline$ $2$ . $9x + 11y = 1528$ $\newline$ And the augmented matrix is: $\newline$ \begin{bmatrix}0 & 1 | & 89\9 & 11 | & 1528\end{bmatrix}
Eliminate Variable $y$ : Since the first equation only contains $y$ , we can use it to eliminate $y$ from the second equation. Multiply the first equation by $-11$ and add it to the second equation: $\newline$ $-11 \times [0 1 | 89] = [-0 -11 | -979]$ $\newline$ $[9 11 | 1528] + [-0 -11 | -979] = [9 0 | 549]$
New System of Equations: Now we have the new system of equations: $\newline$ $1$ . $y = 89$ $\newline$ $2$ . $9x = 549$
Solve for x: Divide the second equation by $9$ to solve for x: $\newline$ $\frac{9x}{9} = \frac{549}{9}$ $\newline$ $x = 61$