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Zorluk: OrtaParallel and Perpendicular Lines

In the standard (x,y)(x, y) coordinate plane, line TT is perpendicular to the line with the equation y=3x−5y = 3x - 5. If line TT passes through the point (6,2)(6, 2), what is the yy-intercept of line TT?

  1. A
    -16
  2. B
    -10
  3. 4Cevap
  4. D
    20
  5. E
    0

Cevap

The correct answer is 4
The slope of the line given by the equation y=3x−5y = 3x - 5 is 33. Since line TT is perpendicular to this line, its slope must be the negative reciprocal of 33, which is −13-\frac{1}{3}. Using the point-slope form with the point (6,2)(6, 2), the equation of line TT is y−2=−13(x−6)y - 2 = -\frac{1}{3}(x - 6). To find the yy-intercept of line TT, we set x=0x = 0 and solve for yy: y−2=−13(0−6)  ⟹  y−2=2  ⟹  y=4y - 2 = -\frac{1}{3}(0 - 6) \implies y - 2 = 2 \implies y = 4.

Adım Adım Çözüm

1
Identify the slope of the given line from its equation.
The slope of the line y=3x−5y = 3x - 5 is 33.
The equation is in slope-intercept form (y=mx+by = mx + b), where mm is the slope.
2
Determine the slope of the perpendicular line TT.
The slope of line TT is −13-\frac{1}{3}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Use the point-slope form to write the equation of line TT with the point (6,2)(6, 2).
The equation of line TT is y−2=−13(x−6)y - 2 = -\frac{1}{3}(x - 6).
The point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1) where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope.
4
Find the yy-intercept of line TT by setting x=0x = 0.
y=4y = 4.
The yy-intercept is the value of yy where the line crosses the yy-axis (when x=0x = 0).

Anahtar Kavram

Finding the equation and y-intercept of a perpendicular line using negative reciprocal slopes
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Konular

Parallel and Perpendicular Lines

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