What is the value of 𝑦 so that 𝐴: negative nine, six; 𝐵: three, negative three; and 𝐶: negative one, 𝑦 are collinear?
The word collinear means that the three points 𝐴, 𝐵, and 𝐶 must all lie on the same straight line. This means that the slope or gradient between each pair of points must be consistent. We can find the slope of the line segment joining two points by finding the change in 𝑦 divided by the change in 𝑥. If we call these two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, then we can use the formula 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.
Now, 𝑦, which is the value we’re looking to find, is the 𝑦-coordinate of the point 𝐶. So this is the approach we’re going to take. First, we’re going to calculate the slope of the line joining the points 𝐴 and 𝐵 as we’ve been given both of these pairs of coordinates. We’ll then find an expression for the slope of the line joining the points 𝐵 and 𝐶. And this expression will be in terms of 𝑦. As we know that the slope must be consistent, this expression must be equal to the value that we find for the slope of the line 𝐴𝐵. This will give us an equation that we can solve to find the value of 𝑦.
So to find the slope of the line joining 𝐴 and 𝐵, first of all, I’m thinking of 𝐴 as the point 𝑥 one, 𝑦 one and 𝐵 as the point 𝑥 two, 𝑦 two. Substituting these values into our formula for the slope gives negative three minus six over three minus negative nine. Now, negative three minus six is negative nine. And in the denominator, those two negatives together make a plus. So we have three plus nine, which is 12. Our slope is negative nine over 12. But we can simplify this by dividing both the numerator and denominator by three. And it gives negative three over four.
So we found the slope of the line joining the points 𝐴 and 𝐵. And now, we want to find an expression for the slope of the line joining the points 𝐵 and 𝐶. Now, this time I’m going to think of 𝐵 as the point 𝑥 one, 𝑦 one and 𝐶 as the point 𝑥 two, 𝑦 two. It doesn’t actually matter which way around you allocate these points. You’ll still get the same slope over all. Substituting these values gives 𝑦 minus negative three in the numerator and negative one minus three in the denominator. This all simplifies to give 𝑦 plus three over negative four.
Now, remember, we said that, for these points to be collinear, the slope must be consistent. So we have that the slope of 𝐴𝐵 must be equal to the slope of 𝐵𝐶. Substituting the two slopes that we found gives negative three over four is equal to 𝑦 plus three over negative four. And this is the equation that we can solve to find the value of 𝑦.
We begin by multiplying both sides of the equation by negative four as this will eliminate the denominator on the right of the equation. But we have to do the same thing to both sides. So on the left of the equation, we have negative three over four multiplied by negative four. We can think of negative four as negative four over one. So we have negative three over four multiplied by negative four over one. Now remember, to multiply fractions, we multiply the numerators and we multiply the denominators. Negative three multiplied by negative four gives 12 and four multiplied by one is four. So we have 12 over four, which is just equal to three.
So our equation has become three is equal to 𝑦 plus three. And the next step for solving for 𝑦 is to subtract three from both sides of the equation. This gives 𝑦 is equal to zero. So we can say that the value of 𝑦, which makes the three points 𝐴, 𝐵, and 𝐶 collinear, is zero.
Now, it’s always sensible to check our answers where we can. And in this case, a way to check would be to calculate the slope of the line joining the points 𝐴 and 𝐶 and check that it is equal to negative three-quarters. So using the point 𝐶 which is negative one, zero as 𝑥 two, 𝑦 two and the point 𝐴 as 𝑥 one, 𝑦 one, we have zero minus six over negative one minus negative nine. In the numerator, we have negative six. And in the denominator, negative one minus negative nine becomes negative one plus nine, which is eight.
So the slope of 𝐴𝐶 is negative six over eight. And this can be simplified by dividing both the numerator and denominator by two to give negative three-quarters. As this is the same as the slope of 𝐴𝐵, this tells us that our answer, 𝑦 equals zero, is correct.