Sure, I'd be happy to explain this step by step!
Given the information:
- We know the line passes through the points [tex]\(A(-6, 6)\)[/tex] and [tex]\(B(12, 3)\)[/tex].
- The slope [tex]\(m\)[/tex] of the line is given as [tex]\(-\frac{1}{6}\)[/tex].
We need to determine the y-intercept [tex]\(b\)[/tex] in the slope-intercept form of the equation of the line, which is [tex]\(y = mx + b\)[/tex].
### Step-by-Step Solution:
1. Identify what we know:
- Slope ([tex]\(m\)[/tex]): [tex]\(-\frac{1}{6}\)[/tex]
- Coordinates of point [tex]\(A\)[/tex]: [tex]\( (-6, 6) \)[/tex]
2. Recall the equation of a line in slope-intercept form:
[tex]\[
y = mx + b
\][/tex]
Where:
- [tex]\(y\)[/tex] is the y-coordinate of the point on the line
- [tex]\(m\)[/tex] is the slope of the line
- [tex]\(x\)[/tex] is the x-coordinate of the point on the line
- [tex]\(b\)[/tex] is the y-intercept of the line
3. Substitute the coordinates of point [tex]\(A\)[/tex] and the slope into the equation:
- For point [tex]\(A(-6, 6)\)[/tex]: [tex]\(x = -6\)[/tex] and [tex]\(y = 6\)[/tex]
- Slope [tex]\(m = -\frac{1}{6}\)[/tex]
Substituting these into [tex]\(y = mx + b\)[/tex]:
[tex]\[
6 = -\frac{1}{6} \times (-6) + b
\][/tex]
4. Solve for [tex]\(b\)[/tex]:
- Calculate the product: [tex]\(-\frac{1}{6} \times (-6)\)[/tex]:
[tex]\[
-\frac{1}{6} \times -6 = 1
\][/tex]
So the equation now looks like:
[tex]\[
6 = 1 + b
\][/tex]
- Isolate [tex]\(b\)[/tex] by subtracting 1 from both sides:
[tex]\[
6 - 1 = b
\][/tex]
[tex]\[
b = 5
\][/tex]
Therefore, the y-intercept [tex]\(b\)[/tex] is [tex]\(5\)[/tex].
So the value of [tex]\(b\)[/tex] is [tex]\(\boxed{5}\)[/tex].
