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]
