Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Substitute teachers with Norwood School District get paid by the day, although subs with teaching credentials earn a different amount than subs without credentials. Yesterday, $3$ non-credentialed subs and $10$ credentialed subs taught in the district. That cost the district $\$1,107$ . Today, $5$ non-credentialed subs and $10$ credentialed subs taught, receiving $\$1,205$ 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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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 any method, and fill in the blanks.

\newline

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

3

non-credentialed subs and

10

credentialed subs taught in the district. That cost the district

\$1,107

. Today,

5

non-credentialed subs and

10

credentialed subs taught, receiving

\$1,205

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 Rates: Let's denote the daily pay for non-credentialed subs as $x$ dollars and for credentialed subs as $y$ dollars. $\newline$ The first equation comes from the first day's payment information: $3$ non-credentialed subs and $10$ credentialed subs cost the district $\$1,107$ . $\newline$ $3x + 10y = 1107$
Formulate Equations: The second equation comes from the second day's payment information: $5$ non-credentialed subs and $10$ credentialed subs received $\$1,205$ from the district. $\newline$ $5x + 10y = 1205$
Solve System of Equations: We now have a system of two equations with two variables: $\newline$ $3x + 10y = 1107$ $\newline$ $5x + 10y = 1205$ $\newline$ We can solve this system using the elimination method by subtracting the first equation from the second. $\newline$ $(5x + 10y) - (3x + 10y) = 1205 - 1107$
Subtract and Calculate $x$ : Perform the subtraction to find the value of $x$ . $5x + 10y - 3x - 10y = 1205 - 1107$ $2x = 98$ $x = \frac{98}{2}$ $x = 49$
Find y Value: Substitute the value of x into one of the original equations to find the value of y. $\newline$ $3(49) + 10y = 1107$ $\newline$ $147 + 10y = 1107$ $\newline$ $10y = 1107 - 147$ $\newline$ $10y = 960$ $\newline$ $y = \frac{960}{10}$ $\newline$ $y = 96$
Final Daily Pay Rates: We have found the values of $x$ and $y$ , which represent the daily pay for non-credentialed and credentialed subs, respectively. $x = 49$ $y = 96$ Subs without credentials get paid $\$49$ per day, and subs with credentials get paid $\$96$ per day.