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

b. neither person makes a valid argument.

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

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 (\(m_1=m_2\)):
• Case 1: If they also have the same \(y\) - intercept (\(b_1 = b_2\)), then the two lines are identical, and the system of equations (which represents the two lines) has infinitely many solutions.
• Case 2: If they have different \(y\) - intercepts (\(b_1

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

Jake assumed that just because two lines have the same slope, they must be the same line. But this is only true when the \(y\) - intercepts are also the same. Erica is correct because she recognized that we need to check the \(y\) - intercepts in addition to the slopes to determine the number of solutions of the system of equations (or the relationship between the two lines).

Answer:

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