Question

Estimate the solution to the system of these linear equations based on the graph: y=-x+6 and y=3x-2

Answer 1

To estimate the solution to the system of equations given by \( y = -x + 6 \) and \( y = 3x - 2 \), we can find the intersection point of the two lines represented by these equations. This intersection point is the solution to the system of equations.

1. **Set the equations equal to each other** since they both equal \( y \):
\[
-x + 6 = 3x - 2
\]

2. **Solve for \( x \)**:
- Add \( x \) to both sides:
\[
6 = 4x - 2
\]
- Add \( 2 \) to both sides:
\[
8 = 4x
\]
- Divide by \( 4 \):
\[
x = 2
\]

3. **Substitute \( x = 2 \) back into one of the original equations** to find \( y \). We'll use \( y = -x + 6 \):
\[
y = -2 + 6 = 4
\]

Thus, the solution to the system of equations is \( (2, 4) \).

You can verify this by substituting \( x = 2 \) into the second equation \( y = 3x - 2 \):
\[
y = 3(2) - 2 = 6 - 2 = 4
\]

So, the intersection point, and therefore the solution to the system of equations, is \( (2, 4) \).