Question

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. (1 point) Responses 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. The statement is true, because the lines are parallel. The statement is false, because the lines have an intersection point.

Answer 1

To determine the validity of the statement about the system of equations given by \(y = -x + 3\) and \(x + y = 8\), let's analyze the two equations:

1. The first equation, \(y = -x + 3\), has a slope of \(-1\) and a y-intercept of \(3\).
2. We can rewrite the second equation \(x + y = 8\) to the slope-intercept form \(y = -x + 8\). Here, the slope is also \(-1\) and the y-intercept is \(8\).

Both lines have the same slope (\(-1\)), indicating that they are parallel. Since parallel lines never intersect, they have no points in common, meaning there is no solution to this system of equations.

Thus, the correct reasoning is:

The statement is true, because the lines are parallel.

This means the correct response is: "The statement is true, because the lines are parallel."