Question

matt and brian were solving a system of equations. they both noticed that the two lines had the same slope. brian said that because each line in the system had the same slope, the two lines had to be parallel, which meant the solution to the system was
o solutions.\ matt disagreed, and said they should also look at the y-intercepts before determining how many solutions there were. who is correct?

a. neither person makes a valid argument.

b. brian is correct. when two lines have the same slope, they are always parallel.

c. matt is correct. though two lines with equal slopes are often parallel, if they have the same y-intercept, they are the same line and have infinitely many solutions.

Explanation:

To determine who is correct, we analyze the properties of linear equations. Two lines with the same slope can either be parallel (distinct lines, no solution) or coincident (the same line, infinite solutions). The key to distinguishing between these cases is the y - intercept. If the y - intercepts are different, the lines are parallel (no solution). If the y - intercepts are the same, the lines are coincident (infinite solutions). Brian assumed that same - slope lines are always parallel (and thus have no solution), but he didn't consider the case where they could be the same line (when y - intercepts are also the same). Matt is correct because he recognized that we need to check the y - intercepts to determine if the lines are parallel (different y - intercepts, no solution) or coincident (same y - intercept, infinite solutions).

Answer:

C. Matt is correct. Though two lines with equal slopes are often parallel, if they have the same y - intercept, they are the same line and have infinitely many solutions.