To find the equation of the line passing through the point [tex]\((2, -4)\)[/tex] that is parallel to the line [tex]\(y = 3x + 2\)[/tex], we can follow these steps:
1. Identify the slope of the given line:
The given line equation is [tex]\(y = 3x + 2\)[/tex]. 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. Thus, the slope [tex]\(m\)[/tex] of the given line is 3.
2. Use the point-slope form of the line equation:
The equation of a line in point-slope form 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. Substituting the given point [tex]\((2, -4)\)[/tex] and the slope [tex]\(m = 3\)[/tex], we get:
[tex]\[
y - (-4) = 3(x - 2)
\][/tex]
3. Simplify the equation:
Start by simplifying the left-hand side of the equation:
[tex]\[
y + 4 = 3(x - 2)
\][/tex]
Next, distribute the slope (3) on the right-hand side:
[tex]\[
y + 4 = 3x - 6
\][/tex]
4. Isolate [tex]\(y\)[/tex] to convert to slope-intercept form:
Subtract 4 from both sides of the equation to isolate [tex]\(y\)[/tex] on the left:
[tex]\[
y = 3x - 6 - 4
\][/tex]
Simplify the right-hand side:
[tex]\[
y = 3x - 10
\][/tex]
Thus, the equation of the line passing through the point [tex]\((2, -4)\)[/tex] and parallel to the line [tex]\(y = 3x + 2\)[/tex] is:
[tex]\[
y = 3x - 10
\][/tex]
