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)

• the statement is true, because the lines are parallel.
• the statement is true, because the lines have an intersection point.
• the statement is false, because the lines are parallel.
• the statement is false, because the lines have an intersection point.

Explanation:

Step1: Rewrite the second equation

Rewrite \(x + y = 8\) in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept). Solve for \(y\): \(y=-x + 8\).

Step2: Compare slopes and y - intercepts

The first equation is \(y=-x + 3\), so its slope \(m_1=- 1\) and y - intercept \(b_1 = 3\). The second equation \(y=-x + 8\) has a slope \(m_2=-1\) and y - intercept \(b_2 = 8\). Since \(m_1=m_2\) (the slopes are equal) and \(b_1
eq b_2\) (the y - intercepts are different), the two lines are parallel. Parallel lines never intersect, so a system of linear equations with two parallel lines has no solution. So the statement "Solving the system \(y=-x + 3\) and \(x + y = 8\) by graphing shows that there is no solution to the system" is true because the lines are parallel.

Answer:

The statement is true, because the lines are parallel.