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Math  /  Algebra

QuestionConsider the line y=−87x+1y=-\frac{8}{7} x+1. Find the equation of the line that is perpendicular to this ine and passes through the point (7,3)(7,3). Find the equation of the line that is parallel to this line and passes through the point (7,3)(7,3).
Equation of perpendicular line: □\square
Equation of parallel line: □\square

Studdy Solution

STEP 1

1. The given line has a slope of −87-\frac{8}{7}.
2. A line perpendicular to another line will have a slope that is the negative reciprocal of the original line's slope.
3. A line parallel to another line will have the same slope as the original line.
4. The point (7,3)(7, 3) is used to find the specific equations of the perpendicular and parallel lines.

STEP 2

1. Determine the slope of the perpendicular line.
2. Use the point-slope form to find the equation of the perpendicular line.
3. Determine the slope of the parallel line.
4. Use the point-slope form to find the equation of the parallel line.

STEP 3

The slope of the given line is −87-\frac{8}{7}.
To find the slope of the line perpendicular to it, take the negative reciprocal:
mperpendicular=78 m_{\text{perpendicular}} = \frac{7}{8}

STEP 4

Use the point-slope form of a line equation, which is y−y1=m(x−x1) y - y_1 = m(x - x_1) , where m m is the slope and (x1,y1)(x_1, y_1) is the point the line passes through.
Substitute m=78 m = \frac{7}{8} and the point (7,3)(7, 3):
y−3=78(x−7) y - 3 = \frac{7}{8}(x - 7)

STEP 5

Simplify the equation:
y−3=78x−498 y - 3 = \frac{7}{8}x - \frac{49}{8}
Add 3 to both sides:
y=78x−498+248 y = \frac{7}{8}x - \frac{49}{8} + \frac{24}{8}
y=78x−258 y = \frac{7}{8}x - \frac{25}{8}
Equation of perpendicular line: y=78x−258\boxed{y = \frac{7}{8}x - \frac{25}{8}}

STEP 6

The slope of the parallel line is the same as the original line's slope, which is −87-\frac{8}{7}.

STEP 7

Again, use the point-slope form of a line equation with m=−87 m = -\frac{8}{7} and the point (7,3)(7, 3):
y−3=−87(x−7) y - 3 = -\frac{8}{7}(x - 7)

STEP 8

Simplify the equation:
y−3=−87x+567 y - 3 = -\frac{8}{7}x + \frac{56}{7}
Add 3 to both sides:
y=−87x+8+3 y = -\frac{8}{7}x + 8 + 3
y=−87x+11 y = -\frac{8}{7}x + 11
Equation of parallel line: y=−87x+11\boxed{y = -\frac{8}{7}x + 11}

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