Question

how many solutions does the system of equations below have?
$y = -5x + 8$
$y = -5x - \frac{4}{9}$
no solution
one solution
infinitely many solutions

Explanation:

Step1: Analyze the slopes of the lines

The two equations are in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the first equation \(y=-5x + 8\), the slope \(m_1=-5\) and the y - intercept \(b_1 = 8\). For the second equation \(y=-5x-\frac{4}{9}\), the slope \(m_2=-5\) and the y - intercept \(b_2=-\frac{4}{9}\).

Step2: Determine the relationship between the lines

Since the slopes of the two lines are equal (\(m_1 = m_2=-5\)) and the y - intercepts are different (\(b_1=8
eq b_2 =-\frac{4}{9}\)), the two lines are parallel. Parallel lines in a plane never intersect. In the context of a system of linear equations, if the lines are parallel, the system has no solution.

Answer:

no solution