Let's solve the problem step-by-step:
1. Identify the variables:
Let:
- [tex]\( x \)[/tex] be the number of 13 cent stamps.
- [tex]\( y \)[/tex] be the number of 18 cent stamps.
2. Set up the equations:
- The total number of stamps is 42:
[tex]\[ x + y = 42 \][/tex]
- The total cost of the stamps is [tex]$6.66. Convert dollars to cents since the stamp values are in cents. Hence, $[/tex]6.66 is equivalent to 666 cents:
[tex]\[ 0.13x + 0.18y = 6.66 \][/tex]
3. System of equations:
The system representing the problem is:
[tex]\[
\begin{cases}
x + y = 42 \\
0.13x + 0.18y = 6.66
\end{cases}
\][/tex]
4. Solve the system:
We have two equations:
[tex]\[
\begin{cases}
x + y = 42 \\
0.13x + 0.18y = 6.66
\end{cases}
\][/tex]
Let's solve these equations to find [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
5. Isolate one variable:
From the first equation, isolate [tex]\( y \)[/tex]:
[tex]\[ y = 42 - x \][/tex]
6. Substitute into the second equation:
Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[
0.13x + 0.18(42 - x) = 6.66
\][/tex]
7. Simplify the second equation:
[tex]\[
0.13x + 0.18 \cdot 42 - 0.18x = 6.66
\][/tex]
[tex]\[
0.13x + 7.56 - 0.18x = 6.66
\][/tex]
[tex]\[
-0.05x + 7.56 = 6.66
\][/tex]
[tex]\[
-0.05x = 6.66 - 7.56
\][/tex]
[tex]\[
-0.05x = -0.90
\][/tex]
[tex]\[
x = \frac{-0.90}{-0.05}
\][/tex]
[tex]\[
x = 18
\][/tex]
8. Find [tex]\( y \)[/tex]:
Substitute [tex]\( x = 18 \)[/tex] back into the equation [tex]\( y = 42 - x \)[/tex]:
[tex]\[
y = 42 - 18
\][/tex]
[tex]\[
y = 24
\][/tex]
So, the man bought:
- 18 of the 13 cent stamps.
- 24 of the 18 cent stamps.
Thus, the system representing the problem is:
[tex]\[
\begin{cases}
x + y = 42 \\
0.13x + 0.18y = 6.66
\end{cases}.
\][/tex]
