To determine the correct system of equations that can be used to find the cost of each pen ([tex]\( p \)[/tex]), folder ([tex]\( f \)[/tex]), and notebook ([tex]\( n \)[/tex]), we'll look at the information given and match it with the choices provided.
1. John purchased:
- 2 pens ([tex]\( p \)[/tex])
- 3 folders ([tex]\( f \)[/tex])
- 4 notebooks ([tex]\( n \)[/tex])
For a total cost of \[tex]$20.
This gives the equation: \( 2p + 3f + 4n = 20 \).
2. Cindy purchased:
- 0 pens
- 5 folders (\( f \))
- 5 notebooks (\( n \))
For a total cost of \$[/tex]25.
This gives the equation: [tex]\( 5f + 5n = 25 \)[/tex].
3. Isaiah purchased:
- 3 pens ([tex]\( p \)[/tex])
- 1 folder ([tex]\( f \)[/tex])
- 2 notebooks ([tex]\( n \)[/tex])
For a total cost of \$11.
This gives the equation: [tex]\( 3p + f + 2n = 11 \)[/tex].
Now, let's compare these equations with the provided options:
Option A:
[tex]\[
\begin{array}{r}
2 p+3 f+4 n=20 \\
5 f+5 n=25 \\
3 p+f+2 n=11
\end{array}
\][/tex]
This configuration matches our derived equations exactly:
- [tex]\( 2p + 3f + 4n = 20 \)[/tex]
- [tex]\( 5f + 5n = 25 \)[/tex]
- [tex]\( 3p + f + 2n = 11 \)[/tex]
Option B:
[tex]\[
\begin{array}{r}
p+f+n=20 \\
f+n=25 \\
3 p+f+2 n=11
\end{array}
\][/tex]
This configuration does not match our equations. The first equation should equal 20, not the individual items summing to 20. So, this is incorrect.
Option C:
[tex]\[
\begin{array}{r}
2 p+3 f+4 n=25 \\
5 f+5 n=20 \\
3 p+f+2 n=11
\end{array}
\][/tex]
This is also incorrect. The totals for the first and second equations are incorrect; they have been swapped.
Option D:
[tex]\[
\begin{array}{r}
2 p+3 f+4 n=20 \\
5 f+2 n=11 \\
3 n+f+5 n=25
\end{array}
\][/tex]
This configuration is incorrect. The second and third equations do not match any of our derived equations.
Thus, the correct answer is:
A.
[tex]\[
\begin{array}{r}
2 p+3 f+4 n=20 \\
5 f+5 n=25 \\
3 p+f+2 n=11
\end{array}
\][/tex]
