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QuestionFind the equations of the line perpendicular and parallel to y=−98x+1y=-\frac{9}{8} x+1 through the point (8,−3)(8,-3).

Studdy Solution

STEP 1

Assumptions1. The given line is y=−98x+1y=-\frac{9}{8} x+1 . The slope of the given line is −98-\frac{9}{8}
3. The line perpendicular to the given line will have a slope that is the negative reciprocal of the given line's slope4. The line parallel to the given line will have the same slope as the given line5. Both the perpendicular and parallel lines pass through the point (8,−3)(8,-3)

STEP 2

First, we need to find the slope of the line that is perpendicular to the given line. We can do this by taking the negative reciprocal of the given line's slope.
m⊥=−1mm_{\perp} = -\frac{1}{m}

STEP 3

Now, plug in the given slope to calculate the slope of the line that is perpendicular to the given line.
m⊥=−1−98m_{\perp} = -\frac{1}{-\frac{9}{8}}

STEP 4

Calculate the slope of the line that is perpendicular to the given line.
m⊥=−1−98=89m_{\perp} = -\frac{1}{-\frac{9}{8}} = \frac{8}{9}

STEP 5

Now that we have the slope of the line that is perpendicular to the given line, we can find its equation using the point-slope form of a line.
y−y1=m⊥(x−x1)y - y1 = m_{\perp}(x - x1)

STEP 6

Plug in the values for the slope and the point to calculate the equation of the line that is perpendicular to the given line.
y−(−3)=89(x−8)y - (-3) = \frac{8}{9}(x -8)

STEP 7

implify the equation of the line that is perpendicular to the given line.
y+3=9x−649y +3 = \frac{}{9}x - \frac{64}{9}

STEP 8

Rearrange the equation to get it in slope-intercept form.
y=8x−64−3y = \frac{8}{}x - \frac{64}{} -3

STEP 9

implify the equation to get the final equation of the line that is perpendicular to the given line.
y=89x−919y = \frac{8}{9}x - \frac{91}{9}

STEP 10

Now, we need to find the equation of the line that is parallel to the given line. Since parallel lines have the same slope, the slope of this line is the same as the given line's slope, −98-\frac{9}{8}.

STEP 11

We can find the equation of the line that is parallel to the given line using the point-slope form of a line.
y−y=m(x−x)y - y = m(x - x)

STEP 12

Plug in the values for the slope and the point to calculate the equation of the line that is parallel to the given line.
y−(−)=−98(x−8)y - (-) = -\frac{9}{8}(x -8)

STEP 13

implify the equation of the line that is parallel to the given line.
y+3=−98x+9y +3 = -\frac{9}{8}x +9

STEP 14

Rearrange the equation to get it in slope-intercept form.
y=−98x+9−3y = -\frac{9}{8}x +9 -3

STEP 15

implify the equation to get the final equation of the line that is parallel to the given line.
y=−98x+y = -\frac{9}{8}x +The equation of the line that is perpendicular to the given line and passes through the point (8,−3)(8,-3) is y=89x−919y = \frac{8}{9}x - \frac{91}{9}. The equation of the line that is parallel to the given line and passes through the point (8,−3)(8,-3) is y=−98x+y = -\frac{9}{8}x +.

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