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Line $r$ has an equation of $y= \frac{6}{7} x+1$ . Line $s$ includes the point $(7, - 7)$ and is perpendicular to line $r$ . What is the equation of line $s$ ?

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Write an equation for a parallel or perpendicular line

Full solution

Q. Line

r

has an equation of

y= \frac{6}{7} x+1

. Line

s

includes the point

(7, - 7)

and is perpendicular to line

r

. What is the equation of line

s

?

Perpendicular Lines Slopes: Line $s$ is perpendicular to line $r$ . $\newline$ Are their slopes the same or opposite reciprocals? $\newline$ Slopes of perpendicular lines are opposite reciprocals.
Equation of Line r: Equation of line r: $\newline$ $y = \frac{6}{7}x + 1$ $\newline$ Find the slope of line r. $\newline$ Compare $y = \frac{6}{7}x + 1$ with $y = mx + b$ . $\newline$ $m = \frac{6}{7}$ $\newline$ Slope of line r: $\frac{6}{7}$
Slope of Line $s$ : Line $s$ is perpendicular to $r$ . $\newline$ Slope of line $r$ : $\frac{6}{7}$ $\newline$ Find the slope of line $s$ . $\newline$ Opposite reciprocal of $\frac{6}{7}$ is $-\frac{7}{6}$ . $\newline$ Slope of line $s$ : $-\frac{7}{6}$
Finding y-intercept: For line s: $\newline$ Slope $m$ : $-\frac{7}{6}$ $\newline$ Point: $(7, -7)$ $\newline$ Plug these values in $y = mx + b$ and find the y-intercept. $\newline$ $-7 = -\frac{7}{6}(7) + b$ $\newline$ $-7 = -\frac{49}{6} + b$ $\newline$ To find $b$ , add $\frac{49}{6}$ to both sides of the equation. $\newline$ $-7 + \frac{49}{6} = b$ $\newline$ To combine the terms, convert $-7$ to a fraction with a denominator of $-\frac{7}{6}$ $0$ . $\newline$ $-\frac{7}{6}$ $1$ $\newline$ $-\frac{7}{6}$ $2$
Equation of Line s: For line s: $\newline$ Slope $m$ : $-\frac{7}{6}$ $\newline$ y-intercept $b$ : $\frac{7}{6}$ $\newline$ What is the equation of the line s in slope-intercept form? $\newline$ Substitute $-\frac{7}{6}$ for $m$ and $\frac{7}{6}$ for $b$ in $y = mx + b$ . $\newline$ $y = -\frac{7}{6}x + \frac{7}{6}$

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