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Difficulty: HardSlope of a Line

In the standard (x,y)(x, y) coordinate plane, line l1l_1 passes through the points (1,2)(1, 2) and (5,r)(5, r), and line l2l_2 passes through the points (−1,r)(-1, r) and (3,8)(3, 8). If the slope of line l2l_2 is −2-2 times the slope of line l1l_1, what is the value of rr?

  1. A
    66
  2. B
    44
  3. −4-4Answer
  4. D
    −43-\frac{4}{3}
  5. E
    1414

Answer

The value of rr is −4-4.
The correct value of rr is −4-4. The slope of line l1l_1 is m1=r−24m_1 = \frac{r - 2}{4} and the slope of line l2l_2 is m2=8−r4m_2 = \frac{8 - r}{4}. Given that m2=−2m1m_2 = -2m_1, substituting these expressions yields 8−r4=−2(r−24)\frac{8 - r}{4} = -2\left(\frac{r - 2}{4}\right). Multiplying both sides by 44 gives 8−r=−2r+48 - r = -2r + 4. Adding 2r2r to both sides and subtracting 88 from both sides results in r=−4r = -4.

Step-by-Step Solution

1
Find the slope m1m_1 of line l1l_1 using the coordinates (1,2)(1, 2) and (5,r)(5, r).
m1=r−25−1=r−24m_1 = \frac{r - 2}{5 - 1} = \frac{r - 2}{4}
The slope formula is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Find the slope m2m_2 of line l2l_2 using the coordinates (−1,r)(-1, r) and (3,8)(3, 8).
m2=8−r3−(−1)=8−r4m_2 = \frac{8 - r}{3 - (-1)} = \frac{8 - r}{4}
Applying the slope formula to the points on the second line.
3
Set up the equation using the given relationship m2=−2m1m_2 = -2m_1.
8−r4=−2(r−24)\frac{8 - r}{4} = -2\left(\frac{r - 2}{4}\right)
We are given that the slope of line l2l_2 is −2-2 times the slope of line l1l_1.
4
Solve the equation for rr.
8−r=−2(r−2)⇒8−r=−2r+4⇒r=−48 - r = -2(r - 2) \Rightarrow 8 - r = -2r + 4 \Rightarrow r = -4
Multiply both sides by 44 and simplify to solve for rr.

Key Concept

Slope of a Line

Alternative Method

Instead of solving the equation algebraically, you can test the given choices for rr by calculating the slopes m1m_1 and m2m_2 for each option and checking if m2=−2m1m_2 = -2m_1.
Estimated Time:2m 0s
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Topics

Slope of a Line

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