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

Answer:

B.) 3 /4 .

Explanation:

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

3p=4r

3 =  4 / r /p

3 /4  =  r /p

The correct answer is B.)  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.

Answer 2

Answer:

B

Explanation:

If the point (p, r) lies on y = x + b, then if we plug in p for x and r for y, the equation will hold true. So let's try that:

r = p + b

Now, since the point (2p, 5r) lies on the line y = 2x + b, we can plug 2p in for x and 5r in for y:

5r = 2 * 2p + b

5r = 4p + b

Let's substitute p + b in for r:

5r = 4p + b

5 * (p + b) = 4p + b

5p + 5b = 4p + b

p = -4b

We now have p in terms of b. Let's do the same for r, so plug -4b in for p in

r = p + b:

r = p + b

r = -4b + b

r = -3b

We want to find the value of r/p, so let's right this in terms of b:

r/p

-3b/-4b

The b's and negative signs will cancel out, so we're left with 3/4.

The answer is B.

Hope this helps!