Question

how many solutions does the system of equations below have?
y = x + \frac{1}{2}
y = x - \frac{6}{7}
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 equation \(y=x+\frac{1}{2}\), the slope \(m_1 = 1\) and the y - intercept \(b_1=\frac{1}{2}\). For the equation \(y=x - \frac{6}{7}\), the slope \(m_2 = 1\) and the y - intercept \(b_2=-\frac{6}{7}\).

Step2: Determine the relationship between the lines

Since the slopes of the two lines are equal (\(m_1=m_2 = 1\)) 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 equations has no solution.

Answer:

no solution