To find the solution to the system of equations:
- \( y = 3x + 2 \)
- \( y = -2x + 12 \)
We can solve the system by substituting values of \( x \) into both equations and finding the corresponding \( y \) values.
Let's complete the table for \( x = -2, -1, 0, 1, 2 \):
Table:
| \( x \) | \( y = 3x + 2 \) | \( y = -2x + 12 \) | |-----------|-------------------|---------------------| | -2 | \( y = 3(-2) + 2 = -6 + 2 = -4 \) | \( y = -2(-2) + 12 = 4 + 12 = 16 \) | | -1 | \( y = 3(-1) + 2 = -3 + 2 = -1 \) | \( y = -2(-1) + 12 = 2 + 12 = 14 \) | | 0 | \( y = 3(0) + 2 = 0 + 2 = 2 \) | \( y = -2(0) + 12 = 0 + 12 = 12 \) | | 1 | \( y = 3(1) + 2 = 3 + 2 = 5 \) | \( y = -2(1) + 12 = -2 + 12 = 10 \) | | 2 | \( y = 3(2) + 2 = 6 + 2 = 8 \) | \( y = -2(2) + 12 = -4 + 12 = 8 \) |
Results:
Now we can summarize the \( y \) values obtained for both equations:
- For \( x = -2 \): \( y = -4 \) and \( y = 16 \)
- For \( x = -1 \): \( y = -1 \) and \( y = 14 \)
- For \( x = 0 \): \( y = 2 \) and \( y = 12 \)
- For \( x = 1 \): \( y = 5 \) and \( y = 10 \)
- For \( x = 2 \): \( y = 8 \) and \( y = 8 \)
Intersection Point:
From the table, we see that the equations yield the same value for \( y \) when \( x = 2 \):
- For \( x = 2 \): \( y = 8 \) (from both equations)
Thus, the ordered pair that is a solution to the system of equations is \((2, 8)\).
Final Answer:
The ordered pair that is a solution to the system of equations is \((2, 8)\).