Question

In a linear relationship two data points are `(9,3)` and `(33,9)`. If the function is `y=mx+b`, we have?

(a) `m=4` and `b=-33`
(b) `m=1/4` and `b=3/4`
(c) `m=4` and `b=-123`
(d) `m=1/4` and `b=24`

Answer 1

Looking at my calculations and your question's structure you have asked for the wrong thing. You need the value of `m` and not `b`

`m=1/4 -> "multiple choice a"`

Explanation:

Let point 1 `->P_1->(x_1,y_1)=(9,3)`
Let point 2 `->P_2->(x_2,y_2)=(33,9)`

gradient `->m=("change in y")/("change in x")` as you read left to right on the graph.

`=>m=(y_2-y_1)/(x_2-x_1)=(9-3)/(33-9) = 6/24 -=(6-:3)/(24-:3) = 2/8 = 1/4`

Answer 2

Answer is (b)

Explanation:

As the two data points given in `(x,f(x))` form are `(9,3)` and `(33,9)`

the linear relation is `(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`, where `y=f(x)`

i.e. `(y-3)/(9-3)=(x-9)/(33-9)`

or `(y-3)/6=(x-9)/24`

or `y-3=(x-9)/4`

or `4y-12=x-9`

or `4y=x-9+12=x+3`

or `y=x/4+3/4`

i.e. `f(x)=1/4xx x+3/4`

Hence, while `m=1/4`, `b=3/4`

and answer is (b)