To solve the problem of determining the number of hot dogs and sodas sold, let's set up and solve a system of linear equations based on the given information.
Let's denote:
- [tex]\( x \)[/tex] as the number of hot dogs sold.
- [tex]\( y \)[/tex] as the number of sodas sold.
Given:
- Each hot dog costs [tex]$\$[/tex]1.5[tex]$.
- Each soda costs $[/tex]\[tex]$0.5$[/tex].
- The total income from sales is [tex]$\$[/tex]78.5$.
- The total number of items (hot dogs and sodas) sold is 87.
We can formulate the following system of equations:
1. The equation representing the total sales in dollars:
[tex]\[ 1.5x + 0.5y = 78.5 \][/tex]
2. The equation representing the total number of items sold:
[tex]\[ x + y = 87 \][/tex]
So our system of equations is:
[tex]\[
\begin{cases}
1.5x + 0.5y = 78.5 \\
x + y = 87
\end{cases}
\][/tex]
### Step-by-Step Solution
1. Multiply the second equation by 0.5 to align the coefficients of [tex]\( y \)[/tex] for elimination:
[tex]\[ 0.5(x + y) = 0.5 \times 87 \][/tex]
[tex]\[ 0.5x + 0.5y = 43.5 \][/tex]
2. Subtract this equation from the first equation to eliminate [tex]\( y \)[/tex]:
[tex]\[ (1.5x + 0.5y) - (0.5x + 0.5y) = 78.5 - 43.5 \][/tex]
[tex]\[ (1.5x - 0.5x) = 35 \][/tex]
[tex]\[ 1x = 35 \][/tex]
[tex]\[ x = 35 \][/tex]
3. Substitute [tex]\( x = 35 \)[/tex] back into the second equation to find [tex]\( y \)[/tex]:
[tex]\[ 35 + y = 87 \][/tex]
[tex]\[ y = 87 - 35 \][/tex]
[tex]\[ y = 52 \][/tex]
Thus, the number of hot dogs sold is 35, and the number of sodas sold is 52.
[tex]\[
\boxed{35 \text{ hot dogs}, 52 \text{ sodas}}
\][/tex]
