Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.

To determine the truth of the statement, let's analyze the two equations given:

1. \( y = -x + 3 \)
2. \( x + y = 8 \)

We can rearrange the second equation in the slope-intercept form (y = mx + b) to compare it with the first equation:

For \( x + y = 8 \):
- Subtract \( x \) from both sides:
\[ y = -x + 8 \]

Now we have two equations:
1. \( y = -x + 3 \)
2. \( y = -x + 8 \)

Both equations have the same slope \(-1\), which means they are parallel lines. However, they have different y-intercepts (\(3\) and \(8\), respectively).

Since parallel lines do not intersect, there is no solution to the system of equations.

Therefore, the statement "Solving the system shows that there is no solution to the system" is true, and the best response is:

**The statement is true, because the lines are parallel.**

are you sure

Yes, I am sure. Let's break it down clearly to confirm:

We have two equations:

1. **First equation:** \( y = -x + 3 \)

This can be interpreted as a line with a slope of \(-1\) and a y-intercept of \(3\).

2. **Second equation:** \( x + y = 8 \)

Rearranging this equation to the slope-intercept form:
\[
y = -x + 8
\]

This line also has a slope of \(-1\) but a different y-intercept of \(8\).

Now, since both lines have the same slope (\(-1\)) but different y-intercepts (one at \(3\) and the other at \(8\)), the two lines are parallel. Parallel lines do not intersect, meaning there is no point \((x, y)\) that would satisfy both equations simultaneously.

Therefore, there is indeed no solution to the system.

The correct statement, as indicated earlier, is:

**The statement is true, because the lines are parallel.**

If you have any further questions or need any clarifications, feel free to ask!