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What is the slope-intercept form of the equation of the line passing through the point
(6,-1) that is perpendicular to the line
2x-3y=8 ?

$38$ . What is the slope-intercept form of the equation of the line passing through the point $(6,-1)$ that is perpendicular to the line $2 x-3 y=8$ ?

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Write an equation for a parallel or perpendicular line

Full solution

Q.

38

. What is the slope-intercept form of the equation of the line passing through the point

(6,-1)

that is perpendicular to the line

2 x-3 y=8

?

Find Slope: Find the slope of the given line $2x - 3y = 8$ . $\newline$ Rewrite in slope-intercept form $(y = mx + b)$ : $\newline$ $2x - 3y = 8$ $\newline$ $-3y = -2x + 8$ $\newline$ $y = \frac{2}{3}x - \frac{8}{3}$ $\newline$ Slope $(m)$ of the given line = $\frac{2}{3}$
Perpendicular Slope: Determine the slope of the line that is perpendicular to the given line. $\newline$ The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. $\newline$ Negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$
Point-Slope Equation: Use the point-slope form to find the equation of the line passing through $(6, -1)$ with slope $-\frac{3}{2}$ . $\newline$ Point-slope form: $y - y_1 = m(x - x_1)$ $\newline$ Plug in $m = -\frac{3}{2}$ , $x_1 = 6$ , $y_1 = -1$ : $\newline$ $y - (-1) = -\frac{3}{2}(x - 6)$ $\newline$ $y + 1 = -\frac{3}{2}x + 9$
Convert to Slope-Intercept: Convert the equation to slope-intercept form $y = mx + b$ . $\newline$ $y = -\frac{3}{2}x + 9 - 1$ $\newline$ $y = -\frac{3}{2}x + 8$

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