To determine the equation of a line parallel to the given line [tex]\(y = 2x + 4\)[/tex] that passes through the point [tex]\((3, -2)\)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
- The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
- From the equation [tex]\(y = 2x + 4\)[/tex], we can see that the slope [tex]\(m\)[/tex] is [tex]\(2\)[/tex].
2. Use the slope for the new line:
- Because parallel lines have the same slope, the slope of our new line will also be [tex]\(2\)[/tex].
3. Use the point-slope form to find the equation:
- The point-slope form of a line is [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.
- Substitute [tex]\(m = 2\)[/tex] and the given point [tex]\((3, -2)\)[/tex]:
[tex]\[
y - (-2) = 2(x - 3)
\][/tex]
4. Simplify the equation:
- First, simplify inside the parentheses:
[tex]\[
y + 2 = 2(x - 3)
\][/tex]
- Distribute the [tex]\(2\)[/tex]:
[tex]\[
y + 2 = 2x - 6
\][/tex]
- Solve for [tex]\(y\)[/tex] by isolating it on one side of the equation:
[tex]\[
y = 2x - 6 - 2
\][/tex]
[tex]\[
y = 2x - 8
\][/tex]
The equation of the line parallel to [tex]\(y = 2x + 4\)[/tex] and passing through the point [tex]\((3, -2)\)[/tex] is [tex]\(y = 2x - 8\)[/tex].
So, the correct answer is [tex]\(y = 2x - 8\)[/tex].
