Question

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

Answer 1

To estimate the solution to the system of linear equations given by \( y = -x + 6 \) and \( y = 3x - 2 \), we should find the point where the two lines intersect.

1. **Identify the equations**:
- The first line, \( y = -x + 6 \), has a y-intercept at (0, 6) and a slope of -1.
- The second line, \( y = 3x - 2 \), has a y-intercept at (0, -2) and a slope of 3.

2. **Graph the lines**:
- For the line \( y = -x + 6 \), if \( x = 0 \), then \( y = 6 \). If \( x = 6 \), then \( y = 0 \) (point (6, 0)). This line goes downwards from left to right.
- For the line \( y = 3x - 2 \), if \( x = 0 \), then \( y = -2 \). If \( x = 2 \), then \( y = 4 \) (point (2, 4)). This line goes upwards from left to right.

3. **Find the intersection**:
- To find where they intersect algebraically, set the equations equal to each other:
\[
-x + 6 = 3x - 2
\]
- Rearranging gives:
\[
6 + 2 = 3x + x
\]
\[
8 = 4x
\]
\[
x = 2
\]
- Substitute \( x = 2 \) into one of the original equations to find \( y \):
\[
y = -2 + 6 = 4
\]

Thus, the solution to the system of equations, or the point of intersection of the two lines, is \( (2, 4) \).