Question

Line one y = -x + 6
Line 2 y= 3x -2

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 given:

1. \( y = -x + 6 \) (Line 1)
2. \( y = 3x - 2 \) (Line 2)

you can visualize these equations as lines on a Cartesian plane.

### Steps to Estimate the Solution:

1. **Graph the first equation**:
- The y-intercept is 6 (point (0, 6)).
- The slope is -1, which means the line will go down one unit for every unit it goes to the right. You can find another point by moving 1 unit to the right (to (1, 5)) and then down 1 unit.

2. **Graph the second equation**:
- The y-intercept is -2 (point (0, -2)).
- The slope is 3, meaning for every unit you move to the right, you go up 3 units. From the y-intercept (0, -2), you can go 1 unit right to (1, 1) (up 3 units).

### Finding the Intersection:
Once both lines are graphed, you will be looking for the point where they intersect. This intersection point is the solution to the system of equations.

### Estimating the Intersection Point:
Graphically estimating from a plot:
- It may be beneficial to use a graphing tool or plot on graph paper to see the exact point.
- Alternatively, you can set the equations equal to find the precise coordinates algebraically.

### Algebraic Method:
Set the two equations equal to each other:
\[
-x + 6 = 3x - 2
\]

Solving for \( x \):
\[
6 + 2 = 3x + x
\]
\[
8 = 4x
\]
\[
x = 2
\]

Now, substitute \( x = 2 \) back into either equation to find \( y \):
Using the first equation:
\[
y = -2 + 6 = 4
\]

### Solution:
The estimated solution to the system of equations is the point \( (2, 4) \). This means the two lines intersect at this point.