To solve the system of equations:
- \( 4h + 4s = 10 \)
- \( 7h + 5s = 15 \)
We can start with the first equation and simplify it. Dividing everything by 4 gives us:
\[ h + s = 2.5 \quad \text{(Equation 1)} \]
Now we can express \( s \) in terms of \( h \):
\[ s = 2.5 - h \]
Next, we will substitute this expression for \( s \) into the second equation:
\[ 7h + 5s = 15 \]
Substituting \( s \):
\[ 7h + 5(2.5 - h) = 15 \]
Distributing \( 5 \):
\[ 7h + 12.5 - 5h = 15 \]
Combine like terms:
\[ (7h - 5h) + 12.5 = 15 \] \[ 2h + 12.5 = 15 \]
Subtract \( 12.5 \) from both sides:
\[ 2h = 15 - 12.5 \] \[ 2h = 2.5 \]
Now, divide by 2:
\[ h = 1.25 \]
Now that we have \( h \), we can substitute it back into the expression we found for \( s \):
\[ s = 2.5 - h = 2.5 - 1.25 = 1.25 \]
Thus, the solution to the system is:
\[ h = 1.25, \quad s = 1.25 \]
In conclusion, the cost of a hot dog is $1.25 and the cost of a soda is $1.25.