Question

tell how many solutions the linear system has
y = -2x + 7
y + 2x = 12
infinite solutions
same slope, same y - intercept
one solution
intersecting lines: different slopes
hint: did you solve the bottom equation for y to put into slope - intercept form?
no solution parallel lines: same slope, different y - intercept

Explanation:

Step1: Rewrite the second equation

The second equation is \( y + 2x = 12 \). Let's rewrite it in slope - intercept form (\( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept). Subtract \( 2x \) from both sides: \( y=-2x + 12 \).

Step2: Compare the two equations

The first equation is \( y=-2x + 7 \), and the rewritten second equation is \( y=-2x + 12 \). For a linear equation \( y = mx + b \), the slope \( m \) of the first equation is \( - 2 \), and the slope of the second equation is also \( -2 \). The y - intercept of the first equation \( b_1 = 7 \), and the y - intercept of the second equation \( b_2=12 \). Since the slopes are the same (\( m_1 = m_2=-2 \)) and the y - intercepts are different (\( b_1
eq b_2 \)), the two lines are parallel. Parallel lines do not intersect, so the system of linear equations has no solution.

Answer:

No solution (because the two lines \( y = - 2x+7 \) and \( y=-2x + 12 \) have the same slope (\( - 2 \)) and different y - intercepts, so they are parallel and do not intersect, meaning the system has no solution).