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.

Explanation:

Step1: Recall parallel line slope

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 \( m=\frac{3}{4} \), and the slope of the line parallel to it is also \( m = \frac{3}{4} \).

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 point - slope form:
\( y - 3=\frac{3}{4}(x - 2) \)

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 \( y=\frac{3}{4}x-\frac{3}{2}+\frac{6}{2} \)
Simplify the right - hand side: \( y=\frac{3}{4}x+\frac{3}{2} \)

Answer:

The equation of the line is \( y=\frac{3}{4}x+\frac{3}{2} \) (or in standard form \( 3x - 4y=-6 \))