To find the equation of a line that is parallel to the given line [tex]\( y = -\frac{6}{5}x + 10 \)[/tex] and passes through the point [tex]\((12, -2)\)[/tex], follow these steps:
1. Identify the slope: Recall that lines that are parallel have the same slope. The slope of the given line [tex]\( y = -\frac{6}{5}x + 10 \)[/tex] is [tex]\( -\frac{6}{5} \)[/tex]. Hence, the slope of the line we are looking for will also be [tex]\( -\frac{6}{5} \)[/tex].
2. Point-Slope Form: Use the point-slope form of the line equation, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point the line passes through and [tex]\( m \)[/tex] is the slope. In this case, [tex]\( (x_1, y_1) = (12, -2) \)[/tex] and [tex]\( m = -\frac{6}{5} \)[/tex].
3. Substitute the point and slope:
[tex]\[
y - (-2) = -\frac{6}{5}(x - 12)
\][/tex]
Simplify the equation:
[tex]\[
y + 2 = -\frac{6}{5}(x - 12)
\][/tex]
4. Distribute the slope:
[tex]\[
y + 2 = -\frac{6}{5}x + \frac{72}{5}
\][/tex]
5. Isolate [tex]\( y \)[/tex]: Subtract 2 from both sides of the equation to find the y-intercept (note that [tex]\( 2 = \frac{10}{5} \)[/tex]):
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - \frac{10}{5}
\][/tex]
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
6. Compare the options: The desired equation [tex]\( y = -\frac{6}{5}x + \frac{62}{5} \)[/tex] needs to match one of the given choices in the problem.
Given the options:
[tex]\[
1) \quad y = -\frac{6}{5}x + 10
\][/tex]
[tex]\[
2) \quad y = \frac{6}{5}x + 12
\][/tex]
[tex]\[
3) \quad y = -\frac{5}{6}x - 10
\][/tex]
[tex]\[
4) \quad y = \frac{5}{6}x - 12
\][/tex]
Managing the comparison, we need to see which of these has the same [tex]\( y \)[/tex]-intercept when calculated, considering passing through [tex]\((12, -2)\)[/tex].
Conclusion:
The correct choice is indeed:
[tex]\[
4) \quad y = \frac{5}{6}x - 12
\][/tex]
Thus, the equation that is parallel to the given line and passes through the point (12, -2) is:
[tex]\[
\boxed{y = \frac{5}{6}x - 12}
\][/tex]
