To determine the equation of the line passing through the point [tex]\((4, 5)\)[/tex] and parallel to the line [tex]\(y = -2x - 2\)[/tex], we can follow these steps:
1. Identify the slope of the parallel line:
A line parallel to another line has the same slope. The given line [tex]\(y = -2x - 2\)[/tex] is in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Here, the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex]. Therefore, the slope of our parallel line is also [tex]\(-2\)[/tex].
2. Use the point-slope form of the line equation:
The point-slope form of the line equation 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. We use the point [tex]\((4, 5)\)[/tex] given in the problem:
[tex]\[
y - 5 = -2(x - 4)
\][/tex]
3. Simplify to slope-intercept form:
To convert the equation to slope-intercept form [tex]\(y = mx + b\)[/tex], we need to simplify the right-hand side:
[tex]\[
y - 5 = -2(x - 4)
\][/tex]
Distribute the [tex]\(-2\)[/tex] to terms inside the parenthesis:
[tex]\[
y - 5 = -2x + 8
\][/tex]
Add 5 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -2x + 8 + 5
\][/tex]
[tex]\[
y = -2x + 13
\][/tex]
Therefore, the equation of the line in slope-intercept form that passes through the point [tex]\((4, 5)\)[/tex] and is parallel to the line [tex]\(y = -2x - 2\)[/tex] is:
[tex]\[
y = -2x + 13
\][/tex]
