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 the equations \( y = -x + 6 \) and \( y = 3x - 2 \), we can graph the two lines and identify the point where they intersect, which represents the solution to the system.

1. **Graphing the first equation**:
- \( y = -x + 6 \) has a y-intercept at 6, so the point (0, 6) is on the graph.
- The slope is -1, meaning for every 1 unit increase in \( x \), \( y \) decreases by 1. Another point can be calculated by taking \( x = 2 \):
\[
y = -2 + 6 = 4 \quad \Rightarrow \text{Point } (2, 4)
\]
- Plotting these points, you can draw the line.

2. **Graphing the second equation**:
- \( y = 3x - 2 \) has a y-intercept at -2, so the point (0, -2) is on the graph.
- The slope is 3, meaning for every 1 unit increase in \( x \), \( y \) increases by 3. For \( x = 1 \):
\[
y = 3(1) - 2 = 1 \quad \Rightarrow \text{Point } (1, 1)
\]
- Plotting these points, you can draw the line.

3. **Finding the intersection**:
- The point of intersection is where both equations are satisfied. Solving the system algebraically can also yield the intersection. Set the equations equal to each other:
\[
-x + 6 = 3x - 2
\]
Rearranging gives:
\[
6 + 2 = 3x + x
\]
\[
8 = 4x
\]
\[
x = 2
\]
- Substitute \( x = 2 \) back into one of the equations to find \( y \):
\[
y = 3(2) - 2 = 6 - 2 = 4
\]
- Thus, the intersection point is \( (2, 4) \).

Based on this analysis, the estimated solution to the system of equations is approximately \( (2, 4) \).