Question

a system of equations has no solution when the slopes of both lines are the ________ and the y-intercepts are ________.

• same; same
• same; different
• different; different
• different; same

Explanation:

To determine when a system of linear equations has no solution, we analyze the slopes and y - intercepts of the lines. A system of linear equations in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept) has no solution when the lines are parallel. Parallel lines have the same slope (\(m_1=m_2\)) because slope determines the steepness and direction of the line. If the slopes are the same and the y - intercepts are different (\(b_1
eq b_2\)), the lines are parallel and will never intersect, so there is no solution. If the slopes are the same and the y - intercepts are the same, the lines are coincident (they lie on top of each other) and have infinitely many solutions. If the slopes are different, the lines will intersect at exactly one point, so there is one solution. So, a system of equations has no solution when the slopes of both lines are the same and the y - intercepts are different.

Answer:

B. same; different