Question

consider the line 5x - 8y = -7. find the equation of the line that is perpendicular to this line and passes through the point (2, 6). find the equation of the line that is parallel to this line and passes through the point (2, 6). note that the aleks graphing calculator may be helpful in checking your answer. equation of perpendicular line: equation of parallel line:

Explanation:

Step1: Rewrite given line to slope-intercept

Rearrange $5x - 8y = -7$ to $y = mx + b$:
$-8y = -5x -7$
$y = \frac{5}{8}x + \frac{7}{8}$
Slope of given line: $m_1 = \frac{5}{8}$

Step2: Find perpendicular line slope

Perpendicular slope is negative reciprocal:
$m_{\perp} = -\frac{8}{5}$

Step3: Find perpendicular line equation

Use point-slope form $y - y_1 = m(x - x_1)$ with $(2,6)$:
$y - 6 = -\frac{8}{5}(x - 2)$
Multiply by 5 to eliminate fraction:
$5(y - 6) = -8(x - 2)$
$5y - 30 = -8x + 16$
$8x + 5y = 46$

Step4: Find parallel line slope

Parallel lines have equal slope:
$m_{\parallel} = \frac{5}{8}$

Step5: Find parallel line equation

Use point-slope form with $(2,6)$:
$y - 6 = \frac{5}{8}(x - 2)$
Multiply by 8 to eliminate fraction:
$8(y - 6) = 5(x - 2)$
$8y - 48 = 5x - 10$
$5x - 8y = -38$

Answer:

Equation of perpendicular line: $8x + 5y = 46$
Equation of parallel line: $5x - 8y = -38$