Question

a system of equations has infinitely many solutions when the slopes of both lines are the ______ and the y-intercepts are the ______. options: same; same, same; different, different; different, different; same

Explanation:

To determine when a system of linear equations has infinitely many solutions, we analyze the properties of the lines (represented by the equations). For two lines in the form \( y = mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept):

• If the slopes (\( m \)) are different, the lines intersect at exactly one point (one solution).
• If the slopes are the same but the y - intercepts (\( b \)) are different, the lines are parallel and never intersect (no solution).
• If the slopes are the same and the y - intercepts are the same, the two equations represent the same line. Every point on one line is also on the other line, so there are infinitely many solutions.

So, a system of equations has infinitely many solutions when the slopes of both lines are the same and the y - intercepts are the same.

Answer:

A. same; same