Point M has coordinates (6, 3p −1) and point N has coordinates (1− p, 2). The gradient of the straight line joining M and N is − 3 and it crosses the y-axis at (0, r ). What is the value of r ?

Answer:

12) H

Any straight line on a graph is governed by the formula: y = mx + c
This means that any point on that line has to follow this formula.
In this formula,
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Y is a function of X (you will understand as we go along to solve this question) M is the gradient (we are given that gradient is -3)
C is the y-intercept (where the line crosses y-axis), in this case we know that line crosses the y-axis at (0, r). We ignore 0, and we only focus on ‘r’ as any point on the y- axis will have 0 for its x value.
We know points M and N are on the same straight line. Let us take point M.
X value = 6, Y value = 3p-1, M = -3, C=r
Now we just substitute these values in the formula y = mx + c
3p-1=-3(6)+r
3p-1=-18+r
r = 3p-1+18=3p+ 17 (equation 1)
We do the same for point N.
X value = 1 – p, Y value = 2, M = -3, C = r
Now we just substitute these values in the formula y = mx + c
2=-3(1 – p) + r
2=-3+3p+r
r=2+3-3p = 5-3p (equation 2)
CB
We have two equations for ‘r’, we can combine them together now.
3p+17=5-3p
6p = -12
p = -2
We now have the value for ‘p’, simply substitute this value in any of the two equations.
r = 5-3p
r = 5-3(-2)
r=5+6=11