Line $p$ has an equation of $y-3= \frac{1}{4} (x+1)$ . Line $q$ is perpendicular to line $p$ and passes through $(3, -6)$ . What is the equation of line $q$ ?

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Algebra 1

Write an equation for a parallel or perpendicular line

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Q. Line

p

has an equation of

y-3= \frac{1}{4} (x+1)

. Line

q

is perpendicular to line

p

and passes through

(3, -6)

. What is the equation of line

q

Convert to Slope-Intercept Form: Convert the equation of line $p$ to slope-intercept form to find its slope. $\newline$ The equation of line $p$ is given as $y-3 = \frac{1}{4} (x+1)$ . To convert it to slope-intercept form $(y = mx + b)$ , we need to distribute the $\frac{1}{4}$ and move the constant term to the other side. $\newline$ $y - 3 = \frac{1}{4} \cdot x + \frac{1}{4}$ $\newline$ $y = \frac{1}{4} \cdot x + \frac{1}{4} + 3$ $\newline$ $y = \frac{1}{4} \cdot x + \frac{13}{4}$ $\newline$ The slope $(m)$ of line $p$ is $\frac{1}{4}$ .
Find Perpendicular Slope: Determine the slope of line $q$ which is perpendicular to line $p$ . Since line $q$ is perpendicular to line $p$ , its slope will be the negative reciprocal of the slope of line $p$ . The slope of line $p$ is $\frac{1}{4}$ , so the negative reciprocal is $-4$ (since the negative reciprocal of $\frac{1}{4}$ is $-1/(\frac{1}{4}) = -4$ ).
Use Point-Slope Form: Use the point-slope form to find the equation of line $q$ . Line $q$ passes through the point $(3, -6)$ and has a slope of $-4$ . 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 a point on the line. Plugging in the slope and the point into the point-slope form, we get: $y - (-6) = -4(x - 3)$ $y + 6 = -4x + 12$
Convert to Slope-Intercept Form: Convert the equation from point-slope form to slope-intercept form. $\newline$ To convert the equation to slope-intercept form $y = mx + b$ , we need to isolate $y$ on one side of the equation. $\newline$ $y + 6 = -4x + 12$ $\newline$ $y = -4x + 12 - 6$ $\newline$ $y = -4x + 6$ $\newline$ The equation of line $q$ in slope-intercept form is $y = -4x + 6$ .