Line $v$ has an equation of $y = -\frac{10}{9}x - 3$ . Perpendicular to line $v$ is line $w$ , which passes through the point $(2,3)$ . What is the equation of line $w$ ? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

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Algebra 1

Write an equation for a parallel or perpendicular line

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Q. Line

v

has an equation of

y = -\frac{10}{9}x - 3

. Perpendicular to line

v

is line

w

, which passes through the point

(2,3)

. What is the equation of line

w

? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Determine slope of line v: Determine the slope of line v. $\newline$ The equation of line v is given as $y = -\frac{10}{9}x - 3$ . The slope ( $m$ ) of a line in the form $y = mx + b$ is the coefficient of $x$ . Therefore, the slope of line v is $-\frac{10}{9}$ .
Find slope of line w: Find the slope of line w. Since line w is perpendicular to line v, its slope will be the negative reciprocal of the slope of line v. The negative reciprocal of $-\frac{10}{9}$ is $\frac{9}{10}$ .
Use point-slope form: Use the point-slope form to find the equation of line $w$ . We have the slope of line $w$ ( $\frac{9}{10}$ ) and a point through which it passes $(2,3)$ . The point-slope form of a line is $(y - y_1) = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point on the line. Plugging in the values, we get $(y - 3) = \frac{9}{10}(x - 2)$ .
Simplify to slope-intercept form: Simplify the equation to slope-intercept form. $\newline$ Starting with $(y - 3) = \frac{9}{10}(x - 2)$ , we distribute the slope on the right side: $\newline$ $y - 3 = \frac{9}{10} \times x - \frac{9}{10} \times 2$ $\newline$ $y - 3 = \frac{9}{10} \times x - \frac{18}{10}$ $\newline$ Now, add $3$ to both sides to solve for $y$ : $\newline$ $y = \frac{9}{10} \times x - \frac{18}{10} + 3$ $\newline$ $y = \frac{9}{10} \times x - \frac{18}{10} + \frac{30}{10}$ $\newline$ $y = \frac{9}{10} \times x + \frac{12}{10}$ $\newline$ Simplify the fraction $\frac{12}{10}$ to $\frac{6}{5}$ : $\newline$ $y - 3 = \frac{9}{10} \times x - \frac{9}{10} \times 2$ $0$