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 ๐ฆ equals ๐ฅ minus 26? Is it (B) three ๐ฆ equals ๐ฅ plus 26? (C) Three ๐ฆ equals ๐ฅ plus 16. Three ๐ฆ equals negative ๐ฅ minus 16. Or (E) three ๐ฆ equals negative ๐ฅ plus 26.
The question tells us to draw the straight line passing through the points five, seven and eight, six. So, letโs begin by doing that as accurately as possible. The point five, seven has an ๐ฅ-coordinate of five and a ๐ฆ-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, letโs recall what we know about the equation of a line in slopeโintercept form. This is of the form ๐ฆ equals ๐๐ฅ plus ๐. When the equation of a line is given in this form, the coefficient of ๐ฅ, the value of ๐, tells us the slope of the line and the value of ๐ gives us the ๐ฆ-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 ๐ is going to be less than zero. We also see that our value of ๐ is somewhere between eight and nine. So, our next job will be to rewrite each of our equations in slopeโintercept form.
To do so, weโll notice that to make ๐ฆ the subject, weโre going to divide every single equation by three. Letโs begin with equation (A) and divide through by three. ๐ฅ divided by three could be written as ๐ฅ over three or one-third ๐ฅ, and negative 26 divided by three is negative twenty-six thirds. First, we notice that the coefficient of ๐ฅ, the slope of this line, is positive. And in fact, the ๐ฆ-intercept is negative. Because itโs 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 letโs now move on to option (B).
This time, dividing by three and we get ๐ฆ equals one-third ๐ฅ plus 26 over three. Once again, we notice that our value of ๐ is positive. So, the graph of the line ๐ฆ equals one-third ๐ฅ plus 26 over three should be sloping upwards. The value of ๐ looks to be roughly accurate, but because the slope is positive, weโre going to disregard option (B). Letโs now look at option (C). Rearranging for ๐ฆ and we get ๐ฆ equals one-third ๐ฅ 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 ๐ฆ equals negative one-third ๐ฅ minus 16 over three. This time we do have a slope thatโs negative. However, our value of ๐ 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 letโs check. Rearranging option (E) and we get ๐ฆ equals negative one-third ๐ฅ plus 26 over three. We have the negative slope weโre expecting, and we have a positive value for our ๐ฆ-intercept. And of course, 26 over three is one-third less than 27 over three, which is nine. And so, the value of our ๐ฆ-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).