To determine 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], we need to follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -2x - 2 \)[/tex]. In slope-intercept form [tex]\( y = mx + b \)[/tex], [tex]\( m \)[/tex] represents the slope. Therefore, the slope of the given line is [tex]\( m = -2 \)[/tex].
2. Recognize that parallel lines have the same slope:
Since parallel lines have identical slopes, the slope of the new line will also be [tex]\( -2 \)[/tex].
3. Use the slope-intercept form [tex]\( y = mx + b \)[/tex]:
We need to find the y-intercept [tex]\( b \)[/tex] of the new line that passes through the point [tex]\((4, 5)\)[/tex]. We have:
- Slope ([tex]\( m \)[/tex]) is [tex]\( -2 \)[/tex]
- Point [tex]\((x, y)\)[/tex] is [tex]\((4, 5)\)[/tex]
4. Substitute the slope and point into the equation:
Substitute [tex]\( m = -2 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( y = 5 \)[/tex] into the slope-intercept form equation:
[tex]\[
y = mx + b
\][/tex]
[tex]\[
5 = -2(4) + b
\][/tex]
5. Solve for [tex]\( b \)[/tex]:
[tex]\[
5 = -8 + b
\][/tex]
Add 8 to both sides to isolate [tex]\( b \)[/tex]:
[tex]\[
5 + 8 = b
\][/tex]
[tex]\[
b = 13
\][/tex]
6. Write the final equation:
The equation of the line in slope-intercept form is:
[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]
