To solve the system of equations using the substitution method, we have the equations:
- \( y = 4x - 2 \)
- \( y = x + 13 \)
Since both equations are expressed in terms of \( y \), we can set them equal to each other:
\[ 4x - 2 = x + 13 \]
Now, let's solve for \( x \):
- Subtract \( x \) from both sides:
\[ 4x - x - 2 = 13 \] \[ 3x - 2 = 13 \]
- Add 2 to both sides:
\[ 3x = 15 \]
- Divide by 3:
\[ x = 5 \]
Now that we have \( x = 5 \), we can substitute this value back into either of the original equations to find \( y \). We'll use the second equation:
\[ y = x + 13 \] \[ y = 5 + 13 = 18 \]
So, the ordered pair \( (x, y) \) is \( (5, 18) \).
Now, let's check which option corresponds to this solution. None of the provided options are \( (5, 18) \). Let's recheck the given options for the closest solution:
A. \( (4, -2) \)
B. \( (0, -2) \)
C. \( (1, 13) \)
D. \( (6, 18) \)
None of the pairs listed are valid solutions to the system.
Thus, the solution for the system \( y = 4x - 2 \) and \( y = x + 13 \) is \( (5, 18) \), which does not match any of the options provided.