Question

1. what is the equation of a line that passes through the point (8, -1) and is parallel to the line that passes through the points (-2, 2) and (6, 6)?

a) ( y = -2x + 2 )
b) ( y = \frac{1}{2}x - 5 )
c) ( y = \frac{1}{2}x + 3 )
d) ( y = 2x - 3 )

Explanation:

Step1: Find the slope of the parallel line

First, we need to find the slope of the line passing through \((-2, 2)\) and \((6, 6)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(-2,2)\) and \((x_2,y_2)=(6,6)\). Then \(m = \frac{6 - 2}{6 - (-2)}=\frac{4}{8}=\frac{1}{2}\). Since parallel lines have the same slope, the slope of the line we want to find (passing through \((8, - 1)\)) is also \(\frac{1}{2}\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(8,-1)\) and \(m = \frac{1}{2}\). Substitute these values into the formula:
\(y-(-1)=\frac{1}{2}(x - 8)\)
Simplify the left - hand side: \(y + 1=\frac{1}{2}(x - 8)\)
Distribute the \(\frac{1}{2}\) on the right - hand side: \(y+1=\frac{1}{2}x-4\)
Subtract 1 from both sides: \(y=\frac{1}{2}x-4 - 1\), so \(y=\frac{1}{2}x-5\)

Answer:

\(y = \frac{1}{2}x-5\) (assuming the option with \(y=\frac{1}{2}x - 5\) is one of the choices, if the options are as partially visible, this should match the correct one)