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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 $s$ ? 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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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

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

Find Perpendicular Slope: Line $r$ 's slope is $\frac{6}{7}$ . Since line $s$ is perpendicular to $r$ , its slope is the negative reciprocal of $\frac{6}{7}$ , which is $-\frac{7}{6}$ .
Calculate Y-Intercept: Using the point $(7, -7)$ and the slope $-\frac{7}{6}$ , plug into the equation $y = mx + b$ to find $b$ , the y-intercept of line $s$ . $\newline$ $-7 = \left(-\frac{7}{6}\right)\cdot7 + b$
Simplify Equation: Simplify the equation: $-7 = -\frac{49}{6} + b$ .
Solve for b: Add $\frac{49}{6}$ to both sides to solve for b: $b = -7 + \frac{49}{6}$ .
Convert to Fraction: Convert $-7$ to a fraction with a denominator of $6$ : $-7 = -\frac{42}{6}$ .
Add Fractions: Now add $-\frac{42}{6}$ to $\frac{49}{6}$ to get $b$ : $b = -\frac{42}{6} + \frac{49}{6}$ .
Final Equation: Simplify $b$ : $b = \frac{7}{6}$ .
Final Equation: Simplify $b$ : $b = \frac{7}{6}$ .The equation of line $s$ in slope-intercept form is $y = -\frac{7}{6}x + \frac{7}{6}$ .

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