Question

For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.

Answer 1

Our equation for this table could be an equation in slope-intercept form:
y=mx+b
m=slope of the line:
b=y-intercept.

1) we have to calculate the slope of this line.
We need two points, you can choose two points of this table, for example:
the points (4,6) and (10,15)

m=(y-y₀)/(x-x₀)
m=(15-6)/(10-4)
m=9/6
m=3/2

2) we have to find "b";
We have this equation:
y=3/2 x +b

Now, we choose any point of this table; For example (8,12), now we find "b",
y=3/2 x+b
12=3/2 (8)+b
12=12+b
b=12-12
b=0

Therefore: the equation would be y=3/2 x 

More easy:
if y vary directly with x, then: y=(y₂-y₁)/(x₂-x₁) x
You choose any two points of this table:
(8,12)
(10,15)

y=(15-12)/(10-8) x
y=3/2 x

Answer: y=3/2 x