Question

find k so that the line through (5, -1) and (k, 2) is
a. parallel to 4x + 7y = 14,
b. perpendicular to 9x + 20y = 40

a. k = $-\frac{1}{4}$
(type an integer or a simplified fraction.)
b. k = $\frac{127}{20}$
(type an integer or a simplified fraction.)

Explanation:

Part a:

Step 1: Find the slope of \(4x + 7y = 14\)

Rewrite the equation in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
Starting with \(4x + 7y=14\), we solve for \(y\):
\(7y=-4x + 14\)
\(y=-\frac{4}{7}x + 2\). So the slope of the line \(4x + 7y = 14\) is \(m_1=-\frac{4}{7}\).

Step 2: Find the slope of the line through \((5,-1)\) and \((k,2)\)

The slope formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For the points \((5,-1)\) and \((k,2)\), the slope \(m_2=\frac{2-(-1)}{k - 5}=\frac{3}{k - 5}\).

Step 3: Set the slopes equal (parallel lines have equal slopes)

Since the two lines are parallel, \(m_1=m_2\).
So \(-\frac{4}{7}=\frac{3}{k - 5}\)
Cross - multiply: \(-4(k - 5)=7\times3\)
\(-4k+20 = 21\)
Subtract 20 from both sides: \(-4k=21 - 20=1\)
Divide both sides by \(-4\): \(k=-\frac{1}{4}\)

Part b:

Step 1: Find the slope of \(9x + 20y = 40\)

Rewrite the equation in slope - intercept form.
\(20y=-9x + 40\)
\(y=-\frac{9}{20}x + 2\). The slope of the line \(9x + 20y = 40\) is \(m_3=-\frac{9}{20}\).

Step 2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is \(- 1\). Let the slope of the perpendicular line be \(m_4\). Then \(m_3\times m_4=-1\)
\(-\frac{9}{20}\times m_4=-1\)
\(m_4=\frac{20}{9}\)

Step 3: Find the slope of the line through \((5,-1)\) and \((k,2)\)

Using the slope formula, the slope of the line through \((5,-1)\) and \((k,2)\) is \(m_5=\frac{2-(-1)}{k - 5}=\frac{3}{k - 5}\)

Step 4: Set the slopes equal (perpendicular slope relationship)

Since the line through \((5,-1)\) and \((k,2)\) is perpendicular to \(9x + 20y = 40\), \(m_5=m_4\)
\(\frac{3}{k - 5}=\frac{20}{9}\)
Cross - multiply: \(20(k - 5)=3\times9\)
\(20k-100 = 27\)
Add 100 to both sides: \(20k=27 + 100 = 127\)
Divide both sides by 20: \(k=\frac{127}{20}\)

Answer:

s:
a. \(k = \boldsymbol{-\frac{1}{4}}\)
b. \(k=\boldsymbol{\frac{127}{20}}\)