Question

1. given the linear equation, ( y = \frac{3}{4}x - 2 ), find the following:

a) the equation of a line through the point ( (2, 3) ), parallel to the given line.
b) the equation of a line through the point ( (2, 3) ), perpendicular to the given line.

Explanation:

Part (a)

Step1: Recall slope of parallel lines

Parallel lines have equal slopes. The given line \( y = \frac{3}{4}x - 2 \) is in slope - intercept form \( y=mx + b \), where \( m \) is the slope. So the slope of the given line, \( m=\frac{3}{4} \). The slope of the line parallel to it, \( m_{parallel}=\frac{3}{4} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(2,3) \) and \( m = \frac{3}{4} \). Substitute these values into the formula:
\( y - 3=\frac{3}{4}(x - 2) \)

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y-3=\frac{3}{4}x-\frac{3}{2} \)
Add 3 to both sides. Since \( 3=\frac{6}{2} \), we have \( y=\frac{3}{4}x-\frac{3}{2}+\frac{6}{2}=\frac{3}{4}x+\frac{3}{2} \)

Part (b)

Step1: Recall slope of perpendicular lines

The slope of a line perpendicular to a line with slope \( m \) is the negative reciprocal. If \( m=\frac{3}{4} \), then the slope of the perpendicular line, \( m_{perpendicular}=-\frac{4}{3} \) (because \( m\times m_{perpendicular}=- 1\), so \( m_{perpendicular}=-\frac{1}{m}=-\frac{4}{3} \))

Step2: Use point - slope form

Using the point - slope form \( y - y_1=m(x - x_1) \) with \( (x_1,y_1)=(2,3) \) and \( m = -\frac{4}{3} \)
\( y - 3=-\frac{4}{3}(x - 2) \)

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y - 3=-\frac{4}{3}x+\frac{8}{3} \)
Add 3 to both sides. Since \( 3=\frac{9}{3} \), we get \( y=-\frac{4}{3}x+\frac{8}{3}+\frac{9}{3}=-\frac{4}{3}x+\frac{17}{3} \)

Answer:

s:
a) The equation of the parallel line is \( \boldsymbol{y=\frac{3}{4}x+\frac{3}{2}} \) (or in point - slope form \( y - 3=\frac{3}{4}(x - 2) \))
b) The equation of the perpendicular line is \( \boldsymbol{y = -\frac{4}{3}x+\frac{17}{3}} \) (or in point - slope form \( y - 3=-\frac{4}{3}(x - 2) \))