To determine the equation of the 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 of the given line:
The given line is [tex]\( y = -\frac{6}{5}x + 10 \)[/tex]. The coefficient of [tex]\( x \)[/tex] in this equation is the slope. Therefore, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{6}{5} \)[/tex].
2. Use the point-slope form of a line equation:
The point-slope form of a line equation 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. We know the slope [tex]\( m = -\frac{6}{5} \)[/tex], and the point is [tex]\((12, -2)\)[/tex].
3. Substitute the known values [tex]\( m = -\frac{6}{5} \)[/tex], [tex]\( x_1 = 12 \)[/tex], and [tex]\( y_1 = -2 \)[/tex] into the point-slope form:
[tex]\[
y - (-2) = -\frac{6}{5}(x - 12)
\][/tex]
Simplify the equation:
[tex]\[
y + 2 = -\frac{6}{5}x + \frac{6}{5} \cdot 12
\][/tex]
[tex]\[
y + 2 = -\frac{6}{5}x + \frac{72}{5}
\][/tex]
4. Isolate [tex]\( y \)[/tex] to put the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - 2
\][/tex]
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - \frac{10}{5}
\][/tex]
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
5. Write the final equation:
The equation of the line parallel to the given line and passing through the point [tex]\((12, -2)\)[/tex] is:
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
6. Identify the correct option:
Among the given options, the equation [tex]\( y = -\frac{6}{5}x + 12 \)[/tex] corresponds to the simplified form of our derived equation. Thus, the correct option is:
Answer: [tex]\( y = -\frac{6}{5}x + 12 \)[/tex]
Hence, the correct option is:
[tex]\[ \boxed{y = -\frac{6}{5} x + 12} \][/tex]
