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Difficulty: MediumCoordinate Geometry and Lines

Line LL is defined by the equation 3x−4y=123x - 4y = 12. Line MM is perpendicular to line LL and passes through the point (6,−1)(6, -1). Which of the following statements about line MM must be true? Select all that apply.

  1. Line MM has a yy-intercept of (0,7)(0, 7).Answer
  2. Line MM passes through the point (3,3)(3, 3).Answer
  3. C
    Line MM passes through Quadrant III.
  4. D
    The xx-intercept of line MM is (7,0)(7, 0).
  5. Line MM is parallel to the line given by 4x+3y=104x + 3y = 10.Answer

Answer

The correct statements are that line MM has a yy-intercept of (0,7)(0, 7), passes through the point (3,3)(3, 3), and is parallel to the line given by 4x+3y=104x + 3y = 10.
Line LL has equation y=34x−3y = \frac{3}{4}x - 3, so its slope is 34\frac{3}{4}. The perpendicular line MM has slope −43-\frac{4}{3}. Using point (6,−1)(6, -1), the equation of line MM is y=−43x+7y = -\frac{4}{3}x + 7. The statement claiming a yy-intercept of (0,7)(0, 7) is correct because y=7y = 7 when x=0x = 0. The statement claiming line MM passes through (3,3)(3, 3) is correct because −43(3)+7=3-\frac{4}{3}(3) + 7 = 3. The statement claiming line MM is parallel to 4x+3y=104x + 3y = 10 is correct because 4x+3y=104x + 3y = 10 has a slope of −43-\frac{4}{3}, which matches the slope of line MM.

Step-by-Step Solution

1
Find the slope of line LL.
Convert 3x−4y=123x - 4y = 12 to slope-intercept form: 4y=3x−12  ⟹  y=34x−34y = 3x - 12 \implies y = \frac{3}{4}x - 3. The slope of line LL is mL=34m_L = \frac{3}{4}.
The slope of a linear equation in standard form can be determined by expressing it as y=mx+by = mx + b.
2
Determine the slope and equation of line MM.
Since line MM is perpendicular to line LL, its slope is the negative reciprocal: mM=−43m_M = -\frac{4}{3}. Using point-slope form with (6,−1)(6, -1): y−(−1)=−43(x−6)  ⟹  y+1=−43x+8  ⟹  y=−43x+7y - (-1) = -\frac{4}{3}(x - 6) \implies y + 1 = -\frac{4}{3}x + 8 \implies y = -\frac{4}{3}x + 7.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Evaluate statement regarding the yy-intercept.
Setting x=0x = 0 gives y=7y = 7, so the yy-intercept is (0,7)(0, 7). This statement is true.
The yy-intercept occurs where x=0x = 0.
4
Evaluate statement regarding point (3,3)(3, 3).
Plug in x=3x = 3: y=−43(3)+7=3y = -\frac{4}{3}(3) + 7 = 3. The point (3,3)(3, 3) lies on line MM. This statement is true.
A point lies on a line if its coordinates satisfy the line equation.
5
Evaluate statement regarding Quadrant III passage.
For x<0x < 0, y=−43x+7>7>0y = -\frac{4}{3}x + 7 > 7 > 0 (Quadrant II). For 0≤x≤5.250 \le x \le 5.25, y≥0y \ge 0 (Quadrant I). For x>5.25x > 5.25, y<0y < 0 (Quadrant IV). The line does not enter Quadrant III. This statement is false.
Lines with negative slopes and positive yy-intercepts pass through Quadrants I, II, and IV only.
6
Evaluate statement regarding the xx-intercept.
Setting y=0y = 0: 0=−43x+7  ⟹  x=214=5.250 = -\frac{4}{3}x + 7 \implies x = \frac{21}{4} = 5.25, giving xx-intercept (214,0)(\frac{21}{4}, 0). This statement is false.
The xx-intercept occurs where y=0y = 0.
7
Evaluate statement regarding parallelism to 4x+3y=104x + 3y = 10.
4x+3y=10  ⟹  y=−43x+1034x + 3y = 10 \implies y = -\frac{4}{3}x + \frac{10}{3}. The slope is −43-\frac{4}{3}, matching line MM's slope, with distinct yy-intercepts. This statement is true.
Lines with identical slopes and different yy-intercepts are parallel.

Key Concept

Perpendicular and parallel line slope relationships, point-slope equation derivation, and coordinate plane quadrant navigation.
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Topics

Coordinate Geometry and Lines

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