Sure, let's go through the steps to find the equation of the line passing through the points [tex]\((-6, -1)\)[/tex] and [tex]\( (6, 1) \)[/tex] using the point-slope formula and then convert it to slope-intercept form.
1. Calculate the Slope:
The slope [tex]\(m\)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given points are [tex]\((x_1, y_1) = (-6, -1)\)[/tex] and [tex]\( (x_2, y_2) = (6, 1) \)[/tex]. Substitute these coordinates into the slope formula:
[tex]\[
m = \frac{1 - (-1)}{6 - (-6)} = \frac{1 + 1}{6 + 6} = \frac{2}{12} = \frac{1}{6}
\][/tex]
2. Write the Point-Slope Form:
The point-slope form of a line equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the point [tex]\((-6, -1)\)[/tex] and the calculated slope [tex]\( m = \frac{1}{6} \)[/tex]:
[tex]\[
y - (-1) = \frac{1}{6}(x - (-6))
\][/tex]
This simplifies to:
[tex]\[
y + 1 = \frac{1}{6}(x + 6)
\][/tex]
3. Convert to Slope-Intercept Form:
The slope-intercept form of a line equation is:
[tex]\[
y = mx + b
\][/tex]
Given [tex]\( y + 1 = \frac{1}{6}(x + 6) \)[/tex]:
[tex]\[
y + 1 = \frac{1}{6}x + \frac{1}{6} \cdot 6
\][/tex]
Simplify:
[tex]\[
y + 1 = \frac{1}{6}x + 1
\][/tex]
Subtract 1 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{6}x + 1 - 1
\][/tex]
[tex]\[
y = \frac{1}{6}x
\][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[
y = \frac{1}{6}x - 1
\][/tex]
