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Difficulty: MediumLinear Functions and Graphs

Two lines, L1L_1 and L2L_2, are graphed in the xyxy-plane. Line L1L_1 passes through the points (2,11)(2, 11) and (6,23)(6, 23). If line L2L_2 is perpendicular to line L1L_1 and contains the point (3,10)(3, 10), what is the xx-coordinate of the xx-intercept of line L2L_2?

Answer: 33

Answer

33
To find the xx-coordinate of the xx-intercept of line L2L_2, first determine the slope of line L1L_1 from the given points (2,11)(2, 11) and (6,23)(6, 23) using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}, which results in m1=23−116−2=3m_1 = \frac{23 - 11}{6 - 2} = 3. Since line L2L_2 is perpendicular to line L1L_1, its slope must be the negative reciprocal of 3, which is −13-\frac{1}{3}. Using the point-slope form with the point (3,10)(3, 10), the equation of line L2L_2 is y−10=−13(x−3)y - 10 = -\frac{1}{3}(x - 3), which simplifies to y=−13x+11y = -\frac{1}{3}x + 11. Setting y=0y = 0 to find the xx-intercept yields 0=−13x+110 = -\frac{1}{3}x + 11, which simplifies to x=33x = 33.

Step-by-Step Solution

1
Calculate the slope of line L1L_1 using the points (2,11)(2, 11) and (6,23)(6, 23) with the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
m1=23−116−2=124=3m_1 = \frac{23 - 11}{6 - 2} = \frac{12}{4} = 3
The slope of a line represents its rate of change and is required to find the relationship with perpendicular lines.
2
Find the slope of line L2L_2 which is perpendicular to L1L_1.
m2=−13m_2 = -\frac{1}{3}
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Write the linear equation for line L2L_2 using the point-slope formula y−y1=m(x−x1)y - y_1 = m(x - x_1) with the point (3,10)(3, 10) and slope m2=−13m_2 = -\frac{1}{3}.
y−10=−13(x−3)  ⟹  y=−13x+11y - 10 = -\frac{1}{3}(x - 3) \implies y = -\frac{1}{3}x + 11
Defining the equation of the line allows us to find the coordinates of any of its intercepts.
4
Set y=0y = 0 in the equation for line L2L_2 to find the xx-coordinate of the xx-intercept.
0=−13x+11  ⟹  13x=11  ⟹  x=330 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33
The xx-intercept is the point where the line crosses the xx-axis, which mathematically corresponds to y=0y = 0.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other. The xx-intercept of a linear graph represents the value of xx when y=0y = 0.
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Topics

Linear Functions and Graphs

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