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)ResponsesThe 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.The statement is false, 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 have an intersection point.The statement is false, because the lines are parallel.

Answer 1

To analyze the system of equations:

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

We can rewrite the second equation in slope-intercept form (y = mx + b):

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

Now we can compare the two equations:
1. \( y = -x + 3 \) (slope = -1, y-intercept = 3)
2. \( y = -x + 8 \) (slope = -1, y-intercept = 8)

Since both lines have the same slope (-1) but different y-intercepts, they are parallel lines. Parallel lines will never intersect, which means there is no solution to the system of equations.

Thus, the statement is **true**, and the correct response is:

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