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Math  /  Algebra

QuestionLine jj has an equation of y+6=6(x−1)y+6=6(x-1). Line kk is perpendicular to line jj and passes through (8,−4)(8,-4). What is the equation of line kk ?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers. □\square

Studdy Solution

STEP 1

1. Line j j has an equation of y+6=6(x−1) y + 6 = 6(x - 1) .
2. Line k k is perpendicular to line j j .
3. Line k k passes through the point (8,−4) (8, -4) .
4. We need to find the equation of line k k in slope-intercept form y=mx+b y = mx + b .

STEP 2

1. Convert the equation of line j j to slope-intercept form to find its slope.
2. Determine the slope of line k k using the perpendicular slope relationship.
3. Use the point-slope form to write the equation of line k k .
4. Convert the equation of line k k to slope-intercept form.

STEP 3

Convert the equation of line j j to slope-intercept form.
Start with the given equation:
y+6=6(x−1) y + 6 = 6(x - 1)
Distribute the 6 6 on the right side:
y+6=6x−6 y + 6 = 6x - 6
Subtract 6 6 from both sides to solve for y y :
y=6x−6−6 y = 6x - 6 - 6
y=6x−12 y = 6x - 12
The slope of line j j is 6 6 .

STEP 4

Determine the slope of line k k .
Since line k k is perpendicular to line j j , the slope of line k k is the negative reciprocal of the slope of line j j .
The slope of line j j is 6 6 , so the slope of line k k is:
m=−16 m = -\frac{1}{6}

STEP 5

Use the point-slope form to write the equation of line k k .
The point-slope form is:
y−y1=m(x−x1) y - y_1 = m(x - x_1)
Substitute m=−16 m = -\frac{1}{6} , x1=8 x_1 = 8 , and y1=−4 y_1 = -4 :
y+4=−16(x−8) y + 4 = -\frac{1}{6}(x - 8)

STEP 6

Convert the equation of line k k to slope-intercept form.
Distribute the slope on the right side:
y+4=−16x+86 y + 4 = -\frac{1}{6}x + \frac{8}{6}
Simplify 86 \frac{8}{6} to 43 \frac{4}{3} :
y+4=−16x+43 y + 4 = -\frac{1}{6}x + \frac{4}{3}
Subtract 4 4 from both sides to solve for y y :
y=−16x+43−4 y = -\frac{1}{6}x + \frac{4}{3} - 4
Convert 4 4 to a fraction with a denominator of 3:
y=−16x+43−123 y = -\frac{1}{6}x + \frac{4}{3} - \frac{12}{3}
Combine the fractions:
y=−16x−83 y = -\frac{1}{6}x - \frac{8}{3}
The equation of line k k in slope-intercept form is:
y=−16x−83 y = -\frac{1}{6}x - \frac{8}{3}

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