Question

(2) consider the equations given below.
part 1: identify the slope of each line.
a. ( y = 5x + 2 )
b. ( y = -\frac{3}{2}x + 4 )
c. ( x - y = 9 )
part 2: state whether the slopes are parallel, perpendicular or neither. explain your answer.
( y = 5x - 1 )
( y = \frac{2}{3}x - 7 )
( y = x + 4 )
| geometry | unit 2: coordinate geometry
1

Explanation:

Part 1:

a.

The equation is in the form \(y = mx + b\) (slope - intercept form), where \(m\) is the slope. For \(y = 5x+2\), the slope \(m = 5\).

b.

For \(y=-\frac{3}{2}x + 4\), using the slope - intercept form \(y = mx + b\), the slope \(m=-\frac{3}{2}\).

c.

Rewrite \(x - y=9\) in slope - intercept form.

Step1:

Isolate \(y\).
\(y=x - 9\)

Step2:

Identify the slope.
Using \(y = mx + b\), the slope \(m = 1\).

Part 2:

a.

For \(y = 5x+2\) (slope \(m_1 = 5\)) and \(y = 5x-1\) (slope \(m_2 = 5\)).
Since \(m_1=m_2 = 5\), the lines are parallel.

b.

For \(y=-\frac{3}{2}x + 4\) (slope \(m_1=-\frac{3}{2}\)) and \(y=\frac{2}{3}x-7\) (slope \(m_2=\frac{2}{3}\)).
Calculate \(m_1\times m_2\).
\(m_1\times m_2=-\frac{3}{2}\times\frac{2}{3}=- 1\).
Since \(m_1\times m_2=-1\), the lines are perpendicular.

c.

For \(y = x + 4\) (slope \(m_1 = 1\)) and \(y=x - 9\) (slope \(m_2 = 1\)).
Since \(m_1=m_2 = 1\), the lines are parallel.

Answer:

Part 1:

a. Slope \(=5\)
b. Slope \(=-\frac{3}{2}\)
c. Slope \(=1\)

Part 2:

a. Parallel (because their slopes are equal, \(m_1 = m_2=5\))
b. Perpendicular (because \(m_1\times m_2=-1\), where \(m_1 =-\frac{3}{2}\) and \(m_2=\frac{2}{3}\))
c. Parallel (because their slopes are equal, \(m_1 = m_2 = 1\))