use the image to answer the question.

Question

x + y = 8
y = -x + 3

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.

a. The statement is false, because the lines have an intersection point.
b. The statement is true, because the lines are parallel
c. The statement is true, because the lines have and intersection.
d. The statement is false, because the lines are parallel.

Answer 1

To determine whether the statement is true or false, we can analyze the given equations:

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

Let's rewrite the first equation in slope-intercept form:

1. From \( x + y = 8 \), we can express \( y \): \[ y = -x + 8 \]

Now we have two equations:

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

Both equations have the same slope, which is -1. However, they have different y-intercepts (8 for Equation 1 and 3 for Equation 2). This means the lines are parallel.

Since parallel lines never intersect, there is indeed no solution to the system of equations.

Therefore, the correct answer is:

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