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$The equation for line j can be written as y+5=5(x+5). Line k includes the point (5,3) and is perpendicular to line j. What is the equation of line k ? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.$

The equation for line $j$ can be written as $y+5=5(x+5)$ . Line $k$ includes the point $(5,3)$ and is perpendicular to line $j$ . What is the equation of line $k$ ? $\newline$ 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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Math Problems

Algebra 1

Write an equation for a parallel or perpendicular line

Full solution

Q. The equation for line

j

can be written as

y+5=5(x+5)

. Line

k

includes the point

(5,3)

and is perpendicular to line

j

. What is the equation of line

k

\newline

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

Find Slope of Line j: Determine the slope of line j. $\newline$ The equation of line j is given as $y + 5 = 5(x + 5)$ . To find the slope, we need to rewrite this equation in slope-intercept form ( $y = mx + b$ ), where $m$ is the slope. $\newline$ $y + 5 = 5(x + 5)$ $\newline$ $y + 5 = 5x + 25$ $\newline$ $y = 5x + 25 - 5$ $\newline$ $y = 5x + 20$ $\newline$ The slope of line j is $5$ .
Find Slope of Line $k$ : Determine the slope of line $k$ . Since line $k$ is perpendicular to line $j$ , its slope will be the negative reciprocal of the slope of line $j$ . The slope of line $j$ is $5$ , so the slope of line $k$ is $-\frac{1}{5}$ .
Use Point-Slope Form: Use the point-slope form to find the equation of line $k$ . Line $k$ passes through the point $(5,3)$ and has a slope of $-\frac{1}{5}$ . 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. Using the point $(5,3)$ and the slope $-\frac{1}{5}$ , we get: $y - 3 = -\frac{1}{5}(x - 5)$
Convert to Slope-Intercept Form: Convert the point-slope form to slope-intercept form. $\newline$ To convert the point-slope form to slope-intercept form $y = mx + b$ , we need to distribute the slope and simplify. $\newline$ $y - 3 = -\frac{1}{5}(x - 5)$ $\newline$ $y - 3 = -\frac{1}{5}x + 1$ $\newline$ $y = -\frac{1}{5}x + 1 + 3$ $\newline$ $y = -\frac{1}{5}x + 4$ $\newline$ The equation of line $k$ in slope-intercept form is $y = -\frac{1}{5}x + 4$ .