Question

question 6 of 10
jake and erica were solving a system of equations. they both noticed that the two lines had the same slope. jake said that because each line in the system had the same slope, the two lines had to be the same, which meant there were infinitely as many solutions to the system. erica disagreed, and said they should also look at the y-intercepts before determining how many solutions there were. who is correct?

a. jake is correct. two lines with the same slope must be the same line.

b. neither person makes a valid argument.

c. erica is correct. two lines with equal slopes could be the same line, but only if they have the same y-intercept.

Explanation:

To determine who is correct, we analyze the properties of linear equations. The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

• If two lines have the same slope ($m_1=m_2$) and the same y - intercept ($b_1 = b_2$), then the two lines are identical, and the system of equations (formed by the two lines) has infinitely many solutions.
• If two lines have the same slope ($m_1=m_2$) but different y - intercepts ($b_1

eq b_2$), then the two lines are parallel and will never intersect, meaning the system of equations has no solution.

Jake's statement that two lines with the same slope must be the same line is incorrect because two lines can have the same slope but different y - intercepts (and thus be parallel, not the same line). Erica is correct because we need to check both the slope and the y - intercept to determine if the lines are the same (infinitely many solutions) or parallel (no solution).

Answer:

C. Erica is correct. Two lines with equal slopes could be the same line, but only if they have the same y - intercept.