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. 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. C. Neither person makes a valid argument.
Ask by Adkins Crawford. in the United States
Feb 26,2025
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Answer
Matt is correct. If two lines have the same slope and the same y-intercept, they are the same line and have infinitely many solutions.
Solution
The correct answer is B. Although two lines with equal slopes are often parallel (which would mean no intersection, hence no solutions), if the two lines also have the same y-intercept, they are actually the same line. In that case, the system of equations represents the same line, and there are infinitely many solutions.
Brian was correct about the case when the slopes are equal and the y-intercepts are different, but he overlooked the possibility that the y-intercepts might be the same. Matt correctly pointed out that it’s important to also consider the y-intercepts before concluding the number of solutions.
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Matt is correct! While it's true that two lines with identical slopes are usually parallel, they can also overlap completely if their \( y \)-intercepts are the same. This means they represent the same line, leading to infinitely many solutions. So, keep those \( y \)-intercepts in mind when analyzing systems of equations! As for Brian's reasoning, he only sees part of the picture. If two lines have the same slope but different \( y \)-intercepts, they're indeed parallel and will never meet, resulting in no solutions. The key factor here is recognizing that lines can sometimes coincide!
