A straight path on a coordinate map is modeled by the equation 2x - y = 8. A second path, … — ACT

A straight path on a coordinate map is modeled by the equation $2x - y = 8$ . A second path, which is perpendicular to the first path, passes through the point $(3, 5)$ . What is the $y$ -intercept of the second path?

A
$-1$
B
$\frac{7}{2}$
$\frac{13}{2}$ Cevap
D
$\frac{33}{2}$
E
$\frac{8}{3}$

Cevap

The y-intercept of the second path is 13/2

The y-intercept is 13/2. First, the slope of the given line is found to be 2. The perpendicular line must have a slope of -1/2. Substituting this slope and the point (3, 5) into the slope-intercept form gives the y-intercept of 13/2.

Adım Adım Çözüm

Find the slope of the first path by rewriting the equation in slope-intercept form (

y = mx + b

The equation

2x - y = 8

becomes

y = 2x - 8

, indicating the slope

m_1 = 2

Knowing the slope of the first path allows us to determine the slope of any perpendicular path.

Calculate the slope of the perpendicular path by taking the negative reciprocal of the first path's slope.

The perpendicular slope

m_2 = -\frac{1}{2}

Perpendicular lines have slopes that are negative reciprocals of one another.

Use the point-slope formula with the point

(3, 5)

and the perpendicular slope

-\frac{1}{2}

to write the equation of the second path.

The equation is

y - 5 = -\frac{1}{2}(x - 3)

, which simplifies to

y = -\frac{1}{2}x + \frac{3}{2} + 5

This establishes the linear relationship of the second path containing the point (3, 5).

Find the y-intercept by setting

x = 0

and simplifying.

Setting

x = 0

gives

y = \frac{3}{2} + 5 = \frac{13}{2}

The y-intercept of a line is the y-coordinate where the line crosses the y-axis (when x = 0).

Anahtar Kavram

Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other, and their equations can be solved using point-slope form.

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