Step-by-step answer:
If x and y both go by equal steps, not necessarily equal steps between x and y, the relation is linear
Here, x goes by steps of 6 (1,7,13,19) and y goes by steps of 3 (4,7,10,13), therefore the relation is linear.
To find the equation passing through all the points, we find first the slope, which is steps in y divided by steps in x, or
slope, m = 3/6 = 1/2
Next, we take any point from the data, say, P0=(x0,y0)=(1,4), and substitute in the point-slope form of the equation
y-y0 = m(x-x0)...........................(1)
since x0=1, y0=4, and m=1/2, we get the equation
y-4 = (1/2)*(x-1) .........................(2)
Simplify (2) to get the slope-intercept form of the linear relation:
y = (1/2)x + 7/2 ........................(3)
Finally, we check the results of the y-values for given values of x, using the relation given in equation (3):
y(1) = 4
y(2)=7
y(3)=10
y(4)=13
all of which correspond exactly to original data, so the equation of the linear relation is correct.