what is value of q? what id the value of y? how do you find the gradients of 4 lines?

Question

the line joining the point P(3,-4) to Q(q, 0) has a gradient of two.
the three points X(2,-1), Y(8,y) and Z(11,2) are collinear....they lie on the same straight line.

Answer 1

I realize that this answer is just a tad late, but hopefully it will be helpful to other people with similar questions.

1. What is the value of q?
2. It is given that the gradient, or slope, as it is called where I am from, is 2. That means that every time x increases by 1, y increases by 2. After all, slope = rise/run or (the change in y)/(the change in x)
3. Between points P and Q, y increases by 4. (0 - (-4) = 4)
4. For y to increase by 4, x had to increase by 2 according to the proportion in the first step (y increases by 2 when x increases by 1).
5. q - 3 = 2 --> q = 3 + 2 = 5.
6. The three points X (2,-1), Y (8,y) and Z (11,2) are collinear.
7. For points to be collinear, they have to have the same slope.
8. The slope between X and Z = (the change in y)/(the change in x) = (2 - (-1))/(11 - 2) = 3/9 = 1/3
9. The slope between X and Y = (the change in y)/(the change in x) = (y - (-1))/(8 - 2) = (y + 1)/6
10. The two slopes have to be equal: 1/3 = (y + 1)/6 --> 2 = y + 1 --> y = 2 - 1 = 1.