Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Substitute teachers with Newberg School District get paid by the day, although subs with teaching credentials earn a different amount than subs without credentials. Yesterday, $1$ non-credentialed sub taught in the district. That cost the district $\$65$ . Today, $8$ non-credentialed subs and $4$ credentialed subs taught, receiving $\$864$ 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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Algebra 1

Solve a system of equations using elimination: 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 Newberg School District get paid by the day, although subs with teaching credentials earn a different amount than subs without credentials. Yesterday,

1

non-credentialed sub taught in the district. That cost the district

\$65

. Today,

8

non-credentialed subs and

4

credentialed subs taught, receiving

\$864

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 Daily Pay: Let's call the daily pay for non-credentialed subs $x$ and for credentialed subs $y$ . $\newline$ First situation: $1$ non-credentialed sub for $\$65$ . $\newline$ $1x + 0y = 65$
First Situation: Second situation: $8$ non-credentialed subs and $4$ credentialed subs for $\$864$ . $8x + 4y = 864$
Second Situation: Now we have the system of equations: $\newline$ $1x + 0y = 65$ $\newline$ $8x + 4y = 864$ $\newline$ We'll solve this using an augmented matrix.
Create Augmented Matrix: Write the augmented matrix for the system: $\newline$ \begin{array}{cc|c} 1 & 0 & 65 \ 8 & 4 & 864 \end{array}
Eliminate Variable $x$ : To make calculations easier, let's multiply the first row by $-8$ and add it to the second row to eliminate $x$ from the second equation. $\newline$ $-8 \times [1 0 | 65] = [-8 0 | -520]$ $\newline$ $[8 4 | 864] + [-8 0 | -520] = [0 4 | 344]$
Solve for Variable $y$ : Now the matrix looks like this: $\newline$ $\begin{matrix} 1 & 0 & | & 65 \ 0 & 4 & | & 344 \end{matrix}$ $\newline$ Divide the second row by $4$ to solve for $y$ . $\newline$ $\frac{\begin{matrix} 0 & 4 & | & 344 \end{matrix}}{4} = \begin{matrix} 0 & 1 & | & 86 \end{matrix}$
Substitute to Solve for x: The new matrix is: $\newline$ $\begin{matrix} 1 & 0 & | & 65 \ 0 & 1 & | & 86 \end{matrix}$ $\newline$ This tells us that $y = 86$ .
Substitute to Solve for x: The new matrix is: $\newline$ $\begin{matrix} 1 & 0 & | & 65 \ 0 & 1 & | & 86 \end{matrix}$ $\newline$ This tells us that $y = 86$ .Substitute $y = 86$ back into the first equation to solve for x. $\newline$ $1x + 0(86) = 65$ $\newline$ $x = 65$