Question Video: Determining the Equation of a Line Passing Through Two Points Mathematics • 6th Grade

Video Transcript

By drawing the straight line that passes through the points five, seven and eight, six on a grid, determine which of the following is the equation of this line. Is it (A) three 𝑦 equals 𝑥 minus 26? Is it (B) three 𝑦 equals 𝑥 plus 26? (C) Three 𝑦 equals 𝑥 plus 16. Three 𝑦 equals negative 𝑥 minus 16. Or (E) three 𝑦 equals negative 𝑥 plus 26.

The question tells us to draw the straight line passing through the points five, seven and eight, six. So, let’s begin by doing that as accurately as possible. The point five, seven has an 𝑥-coordinate of five and a 𝑦-coordinate of seven, and the point eight, six lies here. Constructing a single straight line through these points and we have the line in question. Now to determine which of our equations is the equation of this line. To achieve this, let’s recall what we know about the equation of a line in slope–intercept form. This is of the form 𝑦 equals 𝑚𝑥 plus 𝑐. When the equation of a line is given in this form, the coefficient of 𝑥, the value of 𝑚, tells us the slope of the line and the value of 𝑐 gives us the 𝑦-intercept.

Now, in fact, we might notice that our line is sloping downwards left to right. This means it must have a negative slope. The value of 𝑚 is going to be less than zero. We also see that our value of 𝑐 is somewhere between eight and nine. So, our next job will be to rewrite each of our equations in slope–intercept form.

To do so, we’ll notice that to make 𝑦 the subject, we’re going to divide every single equation by three. Let’s begin with equation (A) and divide through by three. 𝑥 divided by three could be written as 𝑥 over three or one-third 𝑥, and negative 26 divided by three is negative twenty-six thirds. First, we notice that the coefficient of 𝑥, the slope of this line, is positive. And in fact, the 𝑦-intercept is negative. Because it’s negative, it absolutely cannot be between eight and nine. And because the slope of this line is positive, we know that this cannot be the equation of the line we have. So, we disregard option (A), and let’s now move on to option (B).

This time, dividing by three and we get 𝑦 equals one-third 𝑥 plus 26 over three. Once again, we notice that our value of 𝑚 is positive. So, the graph of the line 𝑦 equals one-third 𝑥 plus 26 over three should be sloping upwards. The value of 𝑐 looks to be roughly accurate, but because the slope is positive, we’re going to disregard option (B). Let’s now look at option (C). Rearranging for 𝑦 and we get 𝑦 equals one-third 𝑥 plus sixteen-thirds. Once again, the slope of this line is positive, and 16 divided by three is a little over five, so we can disregard this one on both counts.

Next, looking at option (D), option (D) can be rewritten as 𝑦 equals negative one-third 𝑥 minus 16 over three. This time we do have a slope that’s negative. However, our value of 𝑐 is also negative. This means it cannot be between eight and nine, and so we disregard option (D), meaning that our answer must be (E). But let’s check. Rearranging option (E) and we get 𝑦 equals negative one-third 𝑥 plus 26 over three. We have the negative slope we’re expecting, and we have a positive value for our 𝑦-intercept. And of course, 26 over three is one-third less than 27 over three, which is nine. And so, the value of our 𝑦-intercept is likely to be correct too.

Now in fact, we can check our slope in a little more detail. A slope of negative one-third means that for every one square right, we move one-third down. Alternatively, for every three squares right, we move one down. That indeed is the case on our line. So, we know that its slope is negative one-third. And so, we finished. The equation of the line given is option (E).