Video: Identifying the Equation of the Line for a Given Graph

Consider the graph. Which of the following could be the equation of the line? [A] 𝑦 = (1/3)𝑥 + 1/2 [B] 𝑦 = (−1/3)𝑥 + 2 [C] 𝑦 = (1/3)𝑥 − 2 [D] 𝑦 = (−1/3)𝑥 − 2 [D] 𝑦 = (1/3)𝑥 + 2

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Video Transcript

Consider the graph. Which of the following could be the equation of the line?

So within this question, we’re given a graph and we’re given five possibilities for what the equation of this line could be. Now, the key word in this question is “could.” It’s not saying this is the equation of the line; it’s just saying which is a realistic possibility.

From looking at the graph, we don’t have any scales on either the 𝑥- or the 𝑦-axis. So it wouldn’t be possible to determine the equation of the line exactly. Instead, we’re just considering which of these lines do or don’t have the right characteristics to be the equation of this line.

Each of these lines are in slope-intercept form: 𝑦 equals 𝑚𝑥 plus 𝑐. We’ll answer this question by considering whether each line has a slope and 𝑦-intercept that could match up with those in the diagram.

Let’s look at the slope of the line in the diagram first of all. It’s sloping upwards from left to right, which means it is a positive slope. Therefore, the value of 𝑚 for this line must be positive. Let’s look at the 𝑦-intercept, which is this point here. That point is below the 𝑥-axis. And therefore, the 𝑦-intercept must be negative. So we know that in the equation of this line 𝑚 must be greater than zero, but 𝑐 must be less than zero.

Now, let’s look at the five possibilities that we’re given and we’ll look at the signs of both 𝑚 and 𝑐.

In the first equation, both 𝑚 and 𝑐 are positive, which means this can’t possibly be the equation of the line that we’re looking for. For 𝑦 equals negative a third 𝑥 plus two, 𝑚 is negative and 𝑐 is positive. But that doesn’t match up with what we’re looking for. So we have to rule this one out as well. For the line 𝑦 equals a third 𝑥 minus two, the slope 𝑚 is positive; it’s one-third and the 𝑦-intercept 𝑐 is negative; it’s negative two. And this does match up with what we’re looking for. So this equation is a possibility for the line.

For the equation 𝑦 equals negative a third 𝑥 minus two, both 𝑚 and 𝑐 are negative. So this equation does not match up with the line. Finally, for 𝑦 equals one-third 𝑥 plus two, both 𝑚 and 𝑐 are positive this time. And again, this doesn’t match up with the information in the diagram: 𝑐 should be negative. So this equation is ruled out.

Hence, only one of the five equations has a slope and a 𝑦-intercept that could plausibly be the slope and the 𝑦-intercept of the line in the diagram based on their signs.

Therefore, our answer to the problem is that the only equation which could be the equation of the line is 𝑦 equals one-third 𝑥 minus two.

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