Question

tell if the two given lines are parallel, perpendicular, or just intersect.

1. ( y = -\frac{3}{4}x + 12 ) and ( 3y = 4x - 9 )
2. ( -2x + y = -1 ) and ( -6x + 3y = 6 )

circle the correct answer
parallel perpendicular intersect
circle the correct answer
parallel perpendicular intersect

1. ( x - 3y = 3 ) and ( 2x - 3y = 6 )
2. ( y = \frac{1}{2}x - 2 ) and ( 2x + y = -2 )

circle the correct answer
parallel perpendicular intersect
circle the correct answer
parallel perpendicular intersect
use the slope formula to find the slope of the given line. ( m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} )

1. ( overleftrightarrow{ab} ) given ( a(3,-4) ) and ( b(-5,-2) )
2. ( overleftrightarrow{st} ) given ( s(6,2) ) and ( t(2,-4) )
3. graph ( y=\frac{2}{3}x - 1 )
4. graph the line that is parallel to the line in #7 the passes through ( (-1,2) )
5. graph the line that is perpendicular to the line in #7 that passes through ( (3,-2) )

Explanation:

Step1: Find slopes for 1.

For \(y =-\frac{3}{4}x + 12\), slope \(m_1=-\frac{3}{4}\).
For \(3y = 4x-9\), rewrite as \(y=\frac{4}{3}x - 3\), slope \(m_2=\frac{4}{3}\).
Since \(m_1\times m_2=-\frac{3}{4}\times\frac{4}{3}=- 1\), they are perpendicular.

Step2: Find slopes for 2.

For \(-2x + y=-1\), rewrite as \(y = 2x-1\), slope \(m_1 = 2\).
For \(-6x + 3y=6\), rewrite as \(y = 2x + 2\), slope \(m_2=2\).
Since \(m_1 = m_2\), they are parallel.

Step3: Find slopes for 3.

For \(x-3y = 3\), rewrite as \(y=\frac{1}{3}x - 1\), slope \(m_1=\frac{1}{3}\).
For \(2x-3y = 6\), rewrite as \(y=\frac{2}{3}x - 2\), slope \(m_2=\frac{2}{3}\).
Since \(m_1
eq m_2\) and \(m_1\times m_2
eq - 1\), they just intersect.

Step4: Find slopes for 4.

For \(y=\frac{1}{2}x - 2\), slope \(m_1=\frac{1}{2}\).
For \(2x + y=-2\), rewrite as \(y=-2x - 2\), slope \(m_2=-2\).
Since \(m_1\times m_2=\frac{1}{2}\times(-2)=-1\), they are perpendicular.

Step5: Find slope for 5.

Using \(m=\frac{y_2 - y_1}{x_2 - x_1}\), \(A(3,-4)\), \(B(-5,-2)\)
\(m=\frac{-2-(-4)}{-5 - 3}=\frac{-2 + 4}{-8}=\frac{2}{-8}=-\frac{1}{4}\)

Step6: Find slope for 6.

Using \(m=\frac{y_2 - y_1}{x_2 - x_1}\), \(S(6,2)\), \(T(2,-4)\)
\(m=\frac{-4 - 2}{2 - 6}=\frac{-6}{-4}=\frac{3}{2}\)

Answer:

1. Perpendicular
2. Parallel
3. Intersect
4. Perpendicular
5. \(m =-\frac{1}{4}\)
6. \(m=\frac{3}{2}\)