To find the equation of the line that passes through the point [tex]\((4, 5)\)[/tex] and is parallel to the line [tex]\(y = -2x - 2\)[/tex], follow these steps:
1. Identify the slope of the given line:
- The general form of the slope-intercept equation 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 line equation [tex]\(y = -2x - 2\)[/tex], we see that the slope [tex]\(m\)[/tex] is [tex]\(-2\)[/tex]. Thus, any line parallel to this line will also have a slope of [tex]\(-2\)[/tex].
2. Plug in the slope and the coordinates of the given point into the slope-intercept form:
- The slope-intercept form of the equation of a line is [tex]\(y = mx + b\)[/tex].
- We will use the given point [tex]\((4, 5)\)[/tex] to find the y-intercept [tex]\(b\)[/tex].
- Substitute [tex]\(x = 4\)[/tex], [tex]\(y = 5\)[/tex], and [tex]\(m = -2\)[/tex] into the equation:
[tex]\[
5 = -2(4) + b
\][/tex]
3. Solve for the y-intercept [tex]\(b\)[/tex]:
[tex]\[
5 = -2 \cdot 4 + b
\][/tex]
[tex]\[
5 = -8 + b
\][/tex]
[tex]\[
b = 5 + 8
\][/tex]
[tex]\[
b = 13
\][/tex]
4. Write the equation in slope-intercept form:
- Now that we know the slope is [tex]\(-2\)[/tex] and the y-intercept is [tex]\(13\)[/tex], we can write the equation of the line:
[tex]\[
y = -2x + 13
\][/tex]
Thus, the equation of the line 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].
