Question

In the xy-plane, the point (p,r) lies on the line with equation y=x+b, where b is a constant. The point with coordinates (2p,5r) lies on the line with equation y=2x+b. If p≠0, what is the value of
r
p
?

A)
2
5

B)
3
4

C)
4
3

D)
5
2
​

Answer 1

I hope it is helpful for you

Answer 2

Since the point (p,r) lies on the line with equation y=x+b, the point must satisfy the equation. Substituting p for x and r for y in the equation y=x+b gives r=p+b, or b = r−p.

Similarly, since the point (2p,5r) lies on the line with the equation y=2x+b, the point must satisfy the equation. Substituting 2p for x and 5r for y in the equation y=2x+b gives:

5r=2(2p)+b

5r=4p+b

b = 5r−4p.

Next, we can set the two equations equal to b equal to each other and simplify:

b=r−p=5r−4p

3p=4r

Finally, to find

$\frac{r}{p}$

, we need to divide both sides of the equation by p and by 4:

3p=4r

$3 = \frac{4r}{p}$

$\frac{3}{4} = \frac{r}{p}$

The correct answer is B,

$\frac{3}{4}$

If you picked choices A and D, you may have incorrectly formed your answer out of the coefficients in the point (2p,5r). If you picked Choice C, you may have confused r and p.

Note that while this is in the calculator section of the SAT, you absolutely do not need your calculator to solve it!