Question

What is the solution to the system that is created by the equation y = negative x + 6 and the graph shown below?

On a coordinate plane, a line goes through (0, 0) and (4, 2).
(–8, –4)
(–4, –2)
(4, 2)
(6, 3)

Answer 1

To solve the system created by the equation \( y = -x + 6 \) and the graph of the line passing through the points (0, 0) and (4, 2), we need to find the point where the two lines intersect.

First, let's look at the given equation \( y = -x + 6 \). This line has a y-intercept of 6 and a slope of -1.

Now, let's identify the equation of the line depicted in the graph which goes through the points (0, 0) and (4, 2). The slope \( m \) of this line can be calculated as follows:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{4 - 0} = \frac{2}{4} = \frac{1}{2}
\]

The equation of the line can be written in slope-intercept form \( y = mx + b \). Since it passes through (0, 0), the y-intercept \( b \) is 0. Thus, the equation of the line is:

\[
y = \frac{1}{2}x
\]

Now we need to find the point where the two lines intersect:

1. Start with the two equations:
- \( y = -x + 6 \)
- \( y = \frac{1}{2}x \)

2. Set the equations equal to each other to find the intersection point:
\[
-x + 6 = \frac{1}{2}x
\]

3. Rearranging:
\[
6 = x + \frac{1}{2}x
\]
\[
6 = \frac{3}{2}x
\]

4. Solve for \( x \):
\[
x = \frac{6 \cdot 2}{3} = 4
\]

5. Substitute \( x = 4 \) into either equation to find \( y \). Using \( y = \frac{1}{2x} \):
\[
y = \frac{1}{2}(4) = 2
\]

Thus, the intersection point is \( (4, 2) \).

Looking at the options provided, the solution to the system is:

**(4, 2)**.