To find the equation of the line that passes through the point [tex]\((-6, 5)\)[/tex] and has a slope of [tex]\(-\frac{1}{6}\)[/tex], we can follow these steps:
1. Identify the given information:
- Point: [tex]\((-6, 5)\)[/tex]
- Slope: [tex]\(m = -\frac{1}{6}\)[/tex]
2. Determine which form of the line equation to use:
We will use the point-slope form of the line equation, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes, and [tex]\(m\)[/tex] is the slope.
3. Substitute the given point and slope into the point-slope form:
[tex]\[
y - 5 = -\frac{1}{6}(x + 6)
\][/tex]
4. Simplify the equation to get it into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 5 = -\frac{1}{6}(x + 6)
\][/tex]
Distribute the slope [tex]\(-\frac{1}{6}\)[/tex]:
[tex]\[
y - 5 = -\frac{1}{6}x - 1
\][/tex]
5. Solve for [tex]\(y\)[/tex] to find the y-intercept [tex]\(b\)[/tex]:
Add 5 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{1}{6}x - 1 + 5
\][/tex]
Combine like terms:
[tex]\[
y = -\frac{1}{6}x + 4
\][/tex]
6. Write down the final equation:
The equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex] is:
[tex]\[
y = -\frac{1}{6}x + 4
\][/tex]
So, the equation of the line that passes through the point [tex]\((-6, 5)\)[/tex] and has a slope of [tex]\(-\frac{1}{6}\)[/tex] is:
[tex]\[
y = -\frac{1}{6}x + 4
\][/tex]
