Examkin

Question

Difficulty: HardLinear Equations in Two Variables

A line has the equation y=mx+ky = mx + k in the xyxy-plane, where mm and kk are constants. If the line passes through the points (2,p)(2, p) and (p,−4)(p, -4), and has a yy-intercept of (0,3p)(0, 3p) for some constant p>0p > 0, what is the slope, mm, of the line?

  1. −4-4Answer
  2. B
    44
  3. C
    −1-1
  4. D
    11

Answer

−4-4
The correct answer is −4-4. Substituting the yy-intercept (0,3p)(0, 3p) into the slope-intercept form equation y=mx+ky = mx + k gives k=3pk = 3p. Substituting the point (2,p)(2, p) into y=mx+3py = mx + 3p gives p=2m+3pp = 2m + 3p, which simplifies to m=−pm = -p. Then, substituting the point (p,−4)(p, -4) into y=mx+3py = mx + 3p gives −4=mp+3p-4 = mp + 3p. Replacing mm with −p-p results in −4=−p2+3p-4 = -p^2 + 3p, which can be rearranged to the quadratic equation p2−3p−4=0p^2 - 3p - 4 = 0. Factoring this equation yields (p−4)(p+1)=0(p - 4)(p + 1) = 0. Since p>0p > 0, the only valid solution is p=4p = 4. Substituting p=4p = 4 back into the expression for mm gives the slope m=−4m = -4.

Step-by-Step Solution

1
Determine the y-intercept constant kk in terms of pp.
k=3pk = 3p
The equation of the line is y=mx+ky = mx + k. Since the yy-intercept is given as (0,3p)(0, 3p), substituting x=0x = 0 and y=3py = 3p gives k=3pk = 3p.
2
Find the relationship between the slope mm and the parameter pp.
m=−pm = -p
Since the line passes through (2,p)(2, p), we substitute these coordinates into the equation y=mx+3py = mx + 3p to get p=2m+3pp = 2m + 3p. Solving for mm yields 2m=−2p2m = -2p, or m=−pm = -p.
3
Substitute the point (p,−4)(p, -4) into the linear equation and solve the resulting quadratic equation for pp.
p=4p = 4
Substituting the coordinates of (p,−4)(p, -4) into y=mx+3py = mx + 3p gives −4=mp+3p-4 = mp + 3p. Substituting m=−pm = -p gives −4=−p2+3p-4 = -p^2 + 3p, which simplifies to the quadratic equation p2−3p−4=0p^2 - 3p - 4 = 0. Factoring this equation gives (p−4)(p+1)=0(p - 4)(p + 1) = 0. Since p>0p > 0, we must have p=4p = 4.
4
Calculate the slope mm.
m=−4m = -4
Since m=−pm = -p and we found p=4p = 4, the slope is m=−4m = -4.

Key Concept

Determining the slope of a line from given points and a parameter in a linear equation.

Alternative Method

Instead of finding the equation of the line first, the slope mm can be expressed directly using the slope formula between the points (2,p)(2, p) and (p,−4)(p, -4): m=−4−pp−2m = \frac{-4 - p}{p - 2}. Since the slope of the line connecting (0,3p)(0, 3p) and (2,p)(2, p) is also m=p−3p2−0=−pm = \frac{p - 3p}{2 - 0} = -p, we can set these two expressions for mm equal to each other: −p=−4−pp−2-p = \frac{-4 - p}{p - 2}. Multiplying both sides by p−2p - 2 yields −p(p−2)=−4−p  ⟹  −p2+2p=−4−p-p(p - 2) = -4 - p \implies -p^2 + 2p = -4 - p. Rearranging the terms gives the quadratic equation p2−3p−4=0p^2 - 3p - 4 = 0. Factoring gives (p−4)(p+1)=0(p - 4)(p + 1) = 0. Since p>0p > 0, we find p=4p = 4, and the slope is m=−p=−4m = -p = -4.
Estimated Time:2m 0s
Rate this question

Topics

Linear Equations in Two Variables

More questions from this topic