+129852 When x changes by 3, y changes by (13 - a)
From 4 to 13 x changes by 9 so y changes by 3 (13 - a)= 39 - 3a
And notice that
39 - 3a = b rearrange as
3a + b = 39
The function, g or k ? is linear, so its graph will be a straight line, in which case the gradient between x = 1 and x = 4 will be the same as that between x = 4 and x = 13.
So,
\(\displaystyle \frac{b-13}{13-4}=\frac{13-a}{4-1}.\)
From that,
\(\displaystyle 3b-39=117-9a,\\ 9a+3b=156, \\ 3a + b = 52.\)
Alternatively, let the equation of the function be \(\displaystyle f(x) = Ax + B.\)
Substituting the co-ordinates of the three points, we have
\(\displaystyle a = A+B, \dots(1) \\13=4A+B, \dots(2) \\ b = 13A+B, \dots (3)\)
from which,
\(\displaystyle 3a+b = 3(A+B)+(13A+B)=16A+4B=4(4A+B)=52.\)
