### Equations of Parallel and Perpendicular Lines

#### Writing the Equation of a Line Parallel to a Given Line

Find the equation of the line parallel to the line [tex]$y=4x-2$[/tex] that passes through the point [tex]$(-1,5)$[/tex].

1. The slope of [tex][tex]$y=4x-2$[/tex][/tex] is [tex]4[/tex].
2. The slope of a line parallel to [tex]$y=4x-2$[/tex] is [tex]4[/tex].

The equation of the line parallel to [tex]$y=4x-2$[/tex] that passes through the point [tex][tex]$(-1,5)$[/tex][/tex] is [tex]y=4x+9[/tex].



Answer :

To find the equation of a line parallel to the given line [tex]\( y = 4x - 2 \)[/tex] that passes through the point [tex]\((-1, 5)\)[/tex], follow these steps:

1. Identify the slope of the given line:
The given line has the equation [tex]\( y = 4x - 2 \)[/tex]. The slope-intercept form of a line equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. In this case, [tex]\( m = 4 \)[/tex].
[tex]\[ \text{The slope of } y = 4x - 2 \text{ is } 4. \][/tex]

2. Determine the slope of the parallel line:
Lines that are parallel have the same slope. Therefore, the slope of the line parallel to [tex]\( y = 4x - 2 \)[/tex] is also [tex]\( 4 \)[/tex].
[tex]\[ \text{The slope of a line parallel to } y = 4x - 2 \text{ is } 4. \][/tex]

3. Use the point-slope form to find the equation:
The point-slope form of a line equation is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Here, [tex]\((-1, 5)\)[/tex] is the given point and [tex]\( m = 4 \)[/tex]. Plugging in these values:
[tex]\[ y - 5 = 4(x + 1) \][/tex]

4. Simplify the equation:
Distribute the slope [tex]\(4\)[/tex] on the right-hand side:
[tex]\[ y - 5 = 4x + 4 \][/tex]
Add [tex]\(5\)[/tex] to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 4x + 4 + 5 \][/tex]
Simplify the right-hand side:
[tex]\[ y = 4x + 9 \][/tex]

Thus, the equation of the line parallel to [tex]\( y = 4x - 2 \)[/tex] that passes through the point [tex]\((-1, 5)\)[/tex] is:

[tex]\[ y = 4x + 9 \][/tex]