Question

1. write the equation of the line that is parallel to the line ( 5x - 4y = 4 ) and passes through the point ( (-8, 2) ).

Explanation:

Step1: Find the slope of the given line

First, rewrite the equation \(5x - 4y = 4\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).
Starting with \(5x-4y = 4\), we solve for \(y\):
\(-4y=-5x + 4\)
Divide each term by \(-4\): \(y=\frac{5}{4}x - 1\)
The slope of the line \(y=\frac{5}{4}x - 1\) is \(m=\frac{5}{4}\). Since parallel lines have the same slope, the slope of the line we want to find is also \(m = \frac{5}{4}\).

Step2: Use the point - slope form to find the equation of the line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope.
We know that \(m=\frac{5}{4}\) and the point \((x_1,y_1)=(-8,2)\). Substitute these values into the point - slope form:
\(y - 2=\frac{5}{4}(x+8)\)

Step3: Simplify the equation to slope - intercept form (or standard form)

Expand the right - hand side:
\(y - 2=\frac{5}{4}x+10\)
Add 2 to both sides:
\(y=\frac{5}{4}x+10 + 2\)
\(y=\frac{5}{4}x+12\)

If we want to write it in standard form \(Ax+By = C\) (where \(A\), \(B\), and \(C\) are integers and \(A\gt0\)):
Multiply through by 4 to get rid of the fraction:
\(4y = 5x+48\)
Rearrange to get \(5x-4y=-48\)

Answer:

The equation of the line is \(y=\frac{5}{4}x + 12\) (or \(5x-4y=-48\))