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Consider the graph.
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Which of the following could be the equation of the line?
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So within this question, we’re given a graph and we’re given five possibilities for what the equation of this line could be.
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Now, the key word in this question is “could.”
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It’s not saying this is the equation of the line; it’s just saying which is a realistic possibility.
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From looking at the graph, we don’t have any scales on either the 𝑥- or the 𝑦-axis.
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So it wouldn’t be possible to determine the equation of the line exactly.
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Instead, we’re just considering which of these lines do or don’t have the right characteristics to be the equation of this line.
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Each of these lines are in slope-intercept form: 𝑦 equals 𝑚𝑥 plus 𝑐.
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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.
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Let’s look at the slope of the line in the diagram first of all.
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It’s sloping upwards from left to right, which means it is a positive slope.
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Therefore, the value of 𝑚 for this line must be positive.
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Let’s look at the 𝑦-intercept, which is this point here.
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That point is below the 𝑥-axis.
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And therefore, the 𝑦-intercept must be negative.
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So we know that in the equation of this line 𝑚 must be greater than zero, but 𝑐 must be less than zero.
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Now, let’s look at the five possibilities that we’re given and we’ll look at the signs of both 𝑚 and 𝑐.
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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.
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For 𝑦 equals negative a third 𝑥 plus two, 𝑚 is negative and 𝑐 is positive.
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But that doesn’t match up with what we’re looking for.
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So we have to rule this one out as well.
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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.
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And this does match up with what we’re looking for.
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So this equation is a possibility for the line.
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For the equation 𝑦 equals negative a third 𝑥 minus two, both 𝑚 and 𝑐 are negative.
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So this equation does not match up with the line.
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Finally, for 𝑦 equals one-third 𝑥 plus two, both 𝑚 and 𝑐 are positive this time.
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And again, this doesn’t match up with the information in the diagram: 𝑐 should be negative.
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So this equation is ruled out.
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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.
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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.