To find the equation of the line in slope-intercept form that passes through the point [tex]\((3, -2)\)[/tex] and is parallel to the line [tex]\(y = 2x + 4\)[/tex], we can follow these steps:
1. Determine the slope of the line:
The slope-intercept form of a line is given by [tex]\(y = mx + b\)[/tex] where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. The given line is [tex]\(y = 2x + 4\)[/tex]. Here, the slope [tex]\(m\)[/tex] is [tex]\(2\)[/tex].
2. Use the slope for the parallel line:
Since parallel lines have the same slope, the slope of the line we are looking for is also [tex]\(2\)[/tex].
3. Substitute the given point into the slope-intercept form:
We need to find the y-intercept [tex]\(b\)[/tex] for the new line. Use the point [tex]\((3, -2)\)[/tex], which the line passes through. The slope-intercept form for our line is [tex]\(y = 2x + b\)[/tex]. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -2\)[/tex] into the equation:
[tex]\[
-2 = 2(3) + b
\][/tex]
4. Solve for [tex]\(b\)[/tex]:
[tex]\[
-2 = 6 + b
\][/tex]
Subtract [tex]\(6\)[/tex] from both sides to solve for [tex]\(b\)[/tex]:
[tex]\[
b = -2 - 6
\][/tex]
[tex]\[
b = -8
\][/tex]
5. Write the final equation:
Now we know the slope [tex]\(m = 2\)[/tex] and the y-intercept [tex]\(b = -8\)[/tex]. Substituting these values into the slope-intercept form [tex]\(y = mx + b\)[/tex], the equation of the line is:
[tex]\[
y = 2x - 8
\][/tex]
Thus, the equation of the line in slope-intercept form that passes through [tex]\((3, -2)\)[/tex] and is parallel to the line [tex]\(y = 2x + 4\)[/tex] is [tex]\(\boxed{y = 2x - 8}\)[/tex].
