To solve the system of equations using substitution, we have the following two equations:
- \( 2x + 2y = 38 \)
- \( y = x + 3 \)
Step 1: Substitute \( y \) in the first equation
We can use the second equation to substitute \( y \) in the first equation.
Substituting \( y \) in equation 1:
\[ 2x + 2(x + 3) = 38 \]
Step 2: Simplify the equation
Now, we will simplify the left side of the equation:
\[ 2x + 2x + 6 = 38 \]
Combine like terms:
\[ 4x + 6 = 38 \]
Step 3: Solve for \( x \)
Next, subtract 6 from both sides:
\[ 4x = 38 - 6 \] \[ 4x = 32 \]
Now, divide both sides by 4:
\[ x = \frac{32}{4} \] \[ x = 8 \]
Step 4: Substitute \( x \) back to find \( y \)
Now that we have \( x \), we can substitute \( x = 8 \) back into the second equation to find \( y \):
\[ y = x + 3 \] \[ y = 8 + 3 \] \[ y = 11 \]
Step 5: Write the solution
The solution to the system of equations is:
\[ (x, y) = (8, 11) \]
Step 6: Verify the solution
To ensure the solution is correct, we can substitute \( x = 8 \) and \( y = 11 \) back into the original equations.
For the first equation:
\[ 2(8) + 2(11) = 16 + 22 = 38 \quad \text{(True)} \]
For the second equation:
\[ 11 = 8 + 3 \quad \text{(True)} \]
Both equations are satisfied, confirming that the solution is correct.
Thus, the solution is:
\[ \boxed{(8, 11)} \]