To determine the equation of a line that is parallel to the given line and passes through a specific point, we can follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{6}{5} x + 10 \)[/tex]. The coefficient of [tex]\( x \)[/tex] (i.e., [tex]\(-\frac{6}{5}\)[/tex]) is the slope of the line.
2. Use the slope of the given line for the parallel line:
Since the lines are parallel, the slope of the new line will be the same as that of the given line, [tex]\( m = -\frac{6}{5} \)[/tex].
3. Point-slope form of the line:
To find the equation of the parallel line that passes through the point [tex]\((12, -2)\)[/tex], use the point-slope form of a linear equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1) = (12, -2)\)[/tex] and [tex]\( m = -\frac{6}{5} \)[/tex].
4. Substitute the known values into the point-slope form:
[tex]\[
y - (-2) = -\frac{6}{5}(x - 12)
\][/tex]
Simplifying this, we get:
[tex]\[
y + 2 = -\frac{6}{5}(x - 12)
\][/tex]
[tex]\[
y + 2 = -\frac{6}{5}x + \frac{72}{5}
\][/tex]
5. Isolate [tex]\( y \)[/tex]:
To put the equation in slope-intercept form [tex]\( y = mx + b \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - 2
\][/tex]
Convert [tex]\( -2 \)[/tex] into a fraction with a denominator of 5:
[tex]\[
-2 = -\frac{10}{5}
\][/tex]
Substitute into the equation:
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - \frac{10}{5}
\][/tex]
Combine the constant terms:
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
Hence, the equation of the line that is parallel to the given line and passes through the point [tex]\((12, -2)\)[/tex] is:
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
This can be equivalently written in decimal form as:
[tex]\[
y = -1.2x + 12.4
\][/tex]
Out of the provided options, it corresponds to:
[tex]\[
y = -\frac{6}{5}x + 12.4
\][/tex]
Therefore, none of the provided options fit. However, the correct steps and calculations confirm that the correct answer derived from the process should indeed be:
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5} \quad \text{or} \quad y = -1.2x + 12.4.
\][/tex]
