To solve the system of linear equations given by:
\[ 4h + 4s = 10 \quad \text{(1)} \] \[ 7h + 5s = 15 \quad \text{(2)} \]
we can start by simplifying equation (1).
Step 1: Simplify Equation (1)
Divide the entire equation by 4:
\[ h + s = 2.5 \quad \text{(3)} \]
Step 2: Rearrange Equation (3) to isolate s
From equation (3), we can express \( s \) in terms of \( h \):
\[ s = 2.5 - h \quad \text{(4)} \]
Step 3: Substitute (4) into Equation (2)
Now substitute \( s \) from equation (4) into equation (2):
\[ 7h + 5(2.5 - h) = 15 \]
Distributing the 5:
\[ 7h + 12.5 - 5h = 15 \]
Combine like terms:
\[ 2h + 12.5 = 15 \]
Step 4: Solve for h
Subtract 12.5 from both sides:
\[ 2h = 15 - 12.5 \] \[ 2h = 2.5 \]
Now divide by 2:
\[ h = \frac{2.5}{2} = 1.25 \]
Step 5: Find s using (4)
Now substitute \( h = 1.25 \) back into equation (4):
\[ s = 2.5 - 1.25 \] \[ s = 1.25 \]
Final Answer
Thus, the costs of the hot dog and soda are:
\[ h = 1.25 \] \[ s = 1.25 \]
So the final answers are:
- \( h = $1.25 \)
- \( s = $1.25 \)