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 no solutions. matt disagreed, and said they should also look at the y - intercepts before determining how many solutions there were. who is correct?
a. brian is correct. when two lines have the same slope, they are always parallel.
b. neither person makes a valid argument.
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:

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
If two lines have the same slope \(m\):

• If their \(y\) - intercepts \(b_1

eq b_2\), the lines are parallel and the system of equations (representing the two lines) has no solution.

• If their \(y\) - intercepts \(b_1 = b_2\), the two lines are the same line. In this case, every point on the line is a solution to the system of equations, so there are infinitely many solutions.

Brian only considered the case of parallel lines (different \(y\) - intercepts) when he said "no solutions" just based on the same slope. Matt is correct because he takes into account the \(y\) - intercepts.

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.