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Difficulty: MediumCoordinate Geometry and Lines

In the xyxy-plane, line LL passes through the points (−3,4)(-3, 4) and (5,−2)(5, -2). Line MM is perpendicular to line LL and passes through the point (1,−1)(1, -1). What is the yy-intercept of line MM?

  1. −73-\frac{7}{3}Answer
  2. B
    −74-\frac{7}{4}
  3. C
    −14-\frac{1}{4}
  4. D
    73\frac{7}{3}
  5. E
    13\frac{1}{3}

Answer

The yy-intercept of line MM is −73-\frac{7}{3}.
The slope of line LL is mL=−2−45−(−3)=−34m_L = \frac{-2 - 4}{5 - (-3)} = -\frac{3}{4}. The perpendicular line MM has slope mM=43m_M = \frac{4}{3}. Substituting point (1,−1)(1, -1) into the line equation gives y−(−1)=43(x−1)y - (-1) = \frac{4}{3}(x - 1), which simplifies to y=43x−73y = \frac{4}{3}x - \frac{7}{3}. Thus, the yy-intercept is −73-\frac{7}{3}.

Step-by-Step Solution

1
Calculate the slope of line LL using the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
mL=−2−45−(−3)=−68=−34m_L = \frac{-2 - 4}{5 - (-3)} = \frac{-6}{8} = -\frac{3}{4}.
The slope of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by the vertical change divided by horizontal change.
2
Determine the slope of line MM using the perpendicular line slope relationship.
mM=−1mL=−1−34=43m_M = -\frac{1}{m_L} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3}.
Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.
3
Find the equation of line MM using point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with point (1,−1)(1, -1) and slope mM=43m_M = \frac{4}{3}.
y−(−1)=43(x−1)  ⟹  y+1=43x−43  ⟹  y=43x−73y - (-1) = \frac{4}{3}(x - 1) \implies y + 1 = \frac{4}{3}x - \frac{4}{3} \implies y = \frac{4}{3}x - \frac{7}{3}.
The point-slope form allows writing the linear equation directly given one point and the slope.
4
Identify the yy-intercept of line MM.
The yy-intercept is (0,−73)(0, -\frac{7}{3}), or simply −73-\frac{7}{3}.
In slope-intercept form y=mx+by = mx + b, the constant term bb represents the yy-intercept.

Key Concept

Perpendicular lines have slopes that are negative reciprocals (m1⋅m2=−1m_1 \cdot m_2 = -1). The equation of a line can be determined using its slope and a given point.
Estimated Time:1m 30s
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Topics

Coordinate Geometry and Lines

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