IDNLearn.com: Where your questions meet expert advice and community support. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.

Students are selling hot dogs and sodas. Each hot dog costs [tex]\$ 1.50[/tex] and each soda costs [tex]\$ 0.50[/tex]. They made a total of [tex]\$ 78.50[/tex]. They sold a total of 87 hot dogs and sodas combined. How many hot dogs and how many sodas were sold? Write the system of equations.

[tex]\[
\begin{cases}
1.5x + 0.5y = 78.5 \\
x + y = 87
\end{cases}
\][/tex]


Sagot :

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]