In the 𝑥𝑦xy-plane, the point (𝑝,𝑞)(p,q) lies on the line with equation 𝑦=𝑥+𝑏y=x+b, where 𝑏b is a constant. The point with coordinates (2𝑝,7𝑞)(2p,7q) lies on the line with equation 𝑦=2𝑥+𝑏y=2x+b. If 𝑝p ≠ 00, what is the value of 𝑞𝑝pq ?A2552 B1221 C2772 D7227
Question
In the 𝑥𝑦xy-plane, the point (𝑝,𝑞)(p,q) lies on the line with equation 𝑦=𝑥+𝑏y=x+b, where 𝑏b is a constant. The point with coordinates (2𝑝,7𝑞)(2p,7q) lies on the line with equation 𝑦=2𝑥+𝑏y=2x+b. If 𝑝p ≠ 00, what is the value of 𝑞𝑝pq ?A2552 B1221 C2772 D7227
Solution
Given that the point (p, q) lies on the line y = x + b, we can substitute p for x and q for y in the equation, giving us:
q = p + b ...(1)
Similarly, given that the point (2p, 7q) lies on the line y = 2x + b, we can substitute 2p for x and 7q for y in the equation, giving us:
7q = 2(2p) + b ...(2)
We can simplify equation (2) to:
7q = 4p + b ...(3)
Now we have two equations (1 and 3) with two variables (p and q). We can solve these equations simultaneously to find the values of p and q.
Subtract equation (1) from equation (3) to eliminate b:
7q - q = 4p - p
6q = 3p
Divide both sides by 3p:
q/p = 3/6
Simplify to get:
q/p = 1/2
So, the value of q/p is 1/2. The answer is B) 1/2.
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