macjas wrote:
In the rectangular coordinate system, if the line x = 2y + 5 passes through points (m,n) and (m + 2,n + p), what is the value of p ?
A. -2
B. 0
C. 1/2
D. 1
E. 2
We are given that line x = 2y + 5 passes through points (m,n) and (m + 2,n + p). Thus, we can make two equations and substitute m and m + 2 for x, and n and n + p for y.
Equation 1: The ordered pair (m,n) means that x = m and y = n. Substitute these values into the equation x = 2y + 5.
m = 2n + 5
Equation 2: The ordered pair (m + 2,n + p) means that we will let x = m + 2 and y = n + p.
m + 2 = 2(n + p) + 5
m + 2 = 2n + 2p + 5
m = 2n + 2p + 3
We can equate equations 1 and 2 and we have:
2n + 5 = 2n + 2p + 3
5 = 2p + 3
2 = 2p
p = 1
Alternate solution:
We are given that line x = 2y + 5 passes through points (m,n) and (m + 2,n + p). We can find the slope of the line by isolating y:
x = 2y + 5
2y = x - 5
y = 1/2x - 5/2
Thus, the line has slope 1/2. For any two points on this line, the slope between these two points must be 1/2 also. Since (m,n) and (m + 2,n + p) are on the line, the slope of the line connecting them must be 1/2. Therefore, using the slope formula, which is slope = (change in y)/(change in x), we have:
(n + p - n)/(m + 2 - m) = 1/2
p/2 = 1/2
p = 1
Answer: D