To solve the system of linear equations given by
\[ 4h + 4s = 10 \quad \text{(1)} \] \[ 7h + 5s = 15 \quad \text{(2)} \]
we will first simplify the first equation by dividing through by 4:
\[ h + s = 2.5 \quad \text{(3)} \]
Now we can express \(s\) in terms of \(h\):
\[ s = 2.5 - h \quad \text{(4)} \]
Next, we will substitute equation (4) into equation (2):
\[ 7h + 5(2.5 - h) = 15 \]
Expanding this:
\[ 7h + 12.5 - 5h = 15 \]
Now, combine like terms:
\[ 2h + 12.5 = 15 \]
Subtract 12.5 from both sides:
\[ 2h = 2.5 \]
Now, divide by 2:
\[ h = 1.25 \]
Now that we have the value of \(h\), let's substitute back into equation (4) to find \(s\):
\[ s = 2.5 - 1.25 = 1.25 \]
Thus, the final prices are:
\[ h = 1.25 \quad \text{(cost of a hot dog)} \] \[ s = 1.25 \quad \text{(cost of a soda)} \]
Both prices are \(1.25\).