In the 𝑥𝑦-plane, the line determined by the points three, 𝑚 and 𝑚, 27 passes through the origin. Which of the following could be the value of 𝑚? A) zero, B) nine, C) 18, or D) three.
In this question, we know three points along a line, three 𝑚, 𝑚 27, and the origin, which we know is point zero, zero. These are three locations along the same line. Now, because we don’t know what 𝑚 is, it’s not very easy to sketch this line. Instead, we need to think about some of the things we know about lines. If these three points fall on the same line, then the slope between any of these points will be equal to one another. We also know that we find the slope between two points by finding the change in 𝑦 over the change in 𝑥. And we write that 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. If we label our points 𝑎, 𝑏, and 𝑐, we can look for the slope of the line segment 𝑎𝑏, the slope of line segment 𝑏𝑐, and the slope of line segment 𝑎𝑐.
First, we’ll look at the slope from 𝑎 to 𝑏. To find this, we’ll label point 𝑎: 𝑥 one, 𝑦 one and point 𝑏: 𝑥 two, 𝑦 two. 𝑦 two minus 𝑦 one is 𝑚 minus zero. And 𝑥 two minus 𝑥 one is three minus zero. The slope from point 𝑎 to point 𝑏 is 𝑚 over three. Now, we’ll consider the slope from point 𝑏 to point 𝑐. We’ll label point 𝑏: 𝑥 one, 𝑦 one and point 𝑐: 𝑥 two, 𝑦 two. 𝑦 two minus 𝑦 one is 27 minus 𝑚. And 𝑥 two minus 𝑥 one is 𝑚 minus three. And so, we say the slope from 𝑏 to 𝑐 is 27 minus 𝑚 over 𝑚 minus three. And we can’t simplify that any further.
There is one more slope we can find. We can find the slope from 𝑎 to 𝑐. We’ll let point 𝑎 be 𝑥 one, 𝑦 one and point 𝑐 𝑏e 𝑥 two, 𝑦 two. 𝑦 two minus 𝑦 one is 27 minus zero. And 𝑥 two minus 𝑥 one is 𝑚 minus zero, which is simplified to 27 over 𝑚. Because we know that these points are all on the same line, these three values must be equal to one another. They all represent the slope of this line. We can choose two of these values and set them equal to each other to solve for 𝑚. I chose 𝑚 over three and 27 over 𝑚 because the solving will be a little bit simpler. However, you could still solve with 27 minus 𝑚 over 𝑚 minus three.
To solve here, we cross multiply. Three times 27 and 𝑚 times 𝑚. 𝑚 times 𝑚 equals 𝑚 squared; three times 27 equals 81. We take the square root of both sides, and 𝑚 is equal to positive or negative nine. Looking at our list of answer choices, we only have positive nine. So one possible value of 𝑚 would be nine. Let’s consider the other options.
What about zero? If we plug in zero for 𝑚, it would not be a line. Because we know this line crosses the origin, zero, 27 would also be along the 𝑦-axis. And then, you’d have three, zero which would cross the 𝑥-axis again. There’s no straight line that would pass through these three points. So zero is not an option. If we consider three for the value of 𝑚, a similar thing happens. You have the point zero, zero and the point three, three. But the point three, 27 falls along the same vertical line. There’s no straight line that could pass through all three of these points.
But what about 18? Let’s go back to the ratio of slopes that we found. We know that the slopes between these two lines must be equal and that they must be in the ratio 𝑚 over three is equal to 27 over 𝑚. If we plug 18 in for 𝑚, we have a slope of six on the left and a slope of three-halves on the right. However, when we plug in nine for 𝑚, we have nine over three is equal to 27 over nine, a slope of three on the left and three on the right. Which confirms that nine could be a value for 𝑚 but 18 could not.
But remember, we found two answers, positive nine and negative nine. What would happen if we plugged in negative nine? Well, both sides would have a slope of negative three. Negative nine is still a valid solution for the value of 𝑚. It’s just not one of our answer choices. So we have to go with positive nine.