To determine the system of equations that can be used to find the number of pennies ([tex]\( p \)[/tex]) and quarters ([tex]\( q \)[/tex]), let us analyze the information given:
1. Clara has a total of 28 coins in her purse, consisting of both pennies and quarters.
2. The total value of these coins is 76 cents.
We need two equations to solve for the two unknowns ([tex]\( p \)[/tex] and [tex]\( q \)[/tex]).
Step-by-Step Solution:
1. Equation for Total Coin Count:
Since the total number of coins is 28, we can write the equation:
[tex]\[
p + q = 28
\][/tex]
This equation (Equation 1) states that the sum of the number of pennies ([tex]\( p \)[/tex]) and quarters ([tex]\( q \)[/tex]) is 28.
2. Equation for Total Value:
Next, we need to formulate an equation based on the total value of the coins.
- The value of one penny is 1 cent.
- The value of one quarter is 25 cents.
Given that the total value of all the coins is 76 cents, we can write the equation:
[tex]\[
p + 25q = 76
\][/tex]
This equation (Equation 2) states that the sum of the value of the pennies and the value of the quarters is 76 cents.
Thus, the system of equations that can be used to determine the number of pennies ([tex]\( p \)[/tex]) and quarters ([tex]\( q \)[/tex]) is:
[tex]\[
\begin{cases}
p + q = 28 \\
p + 25q = 76
\end{cases}
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
\begin{array}{l}
p + q = 28 \\
p + 25q = 76
\end{array}
\][/tex]
