Question

Solving the system y=−x+3and x+y=8by graphing shows that there is no solution to the system. Is this statement true or false? Explain.(1 point)ResponsesThe statement is false, because the lines have an intersection point.The statement is false, because the lines have an intersection point.The statement is false, because the lines are parallel.The statement is false, because the lines are parallel.The statement is true, because the lines have an intersection point.The statement is true, because the lines have an intersection point.The statement is true, because the lines are parallel.

Answer 1

To analyze the statement, let's look at the two equations provided:

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

First, we can rewrite the second equation in slope-intercept form (y = mx + b) to make it easier to compare the lines:

Starting from \( x + y = 8 \):
- Subtract \( x \) from both sides to get \( y = -x + 8 \).

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

Now we can identify the slopes and y-intercepts:
- The slope of the first line (from \( y = -x + 3 \)) is \(-1\), and the y-intercept is \(3\).
- The slope of the second line (from \( y = -x + 8 \)) is also \(-1\), and the y-intercept is \(8\).

Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect. Therefore, the system of equations has no solution.

Therefore, the correct response is:
- **The statement is false, because the lines are parallel.**