There are two ways to solve this question:
1. The quickest way to solve this question is perhaps to try substitute an x-value from the table into each of the possible answer equations and seeing if it returns the same y-value as in the table.
Let us use point (0, 2) from the table:
A. y = 12x - 5
if x = 0: y = -5
The value we wanted is 2, therefor A is not correct
B. y = -5x + 2
if x = 0, y = 2
This is correct, however to check if it really is this equation we should substitute a few more points in.
if x = -2: y = -5(-2) + 2 = 12 (correct)
if x = -1: y = -5(-1) + 2 = 7 (correct)
Given that the first three values are correct, we can say that the answer is B. y = -5x + 2.
To make sure even further however we may also substitute x = 0 into C. and D.
C. y = 10x - 5
if x = 0: y = -5 (not correct)
D. y = -2x + 2
if x = 0: y = 2 (this is correct so we must try substituting another value from the table)
if x = -2: y = -2(-2) + 2 = 6 (the value from the table for x = -2 is y = 12, therefor this is not correct)
Thus the answer is B. y = -5x + 2
2. The second method is to find the equation of the line itself without testing points. The first step is to find the gradient. We can do this using any two points, (x1, y1) and (x2, y2), and the formula for the gradient of a straight line:
m = (y2 - y1)/(x2 - x1)
Let's use the first two points from the table (-2, 12) and (-1, 7). Thus:
m = (7 - 12)/(-1 - (-2))
m = (-5)/1
m = -5
Now we can substitute this and one point (let's take (0, 2) into the formula for a straight line y - y1 = m(x - x1), where (x1, y1) is our point. Thus:
y - 2 = -5(x - 0)
y - 2 = -5x
y = -5x + 2
Therefor, looking at the possible answers, we can see that this matches answer B.