To find the equation of a line that is parallel to a given line and passes through a specific point, we can follow a step-by-step approach. Below are the steps and the detailed solution:
1. Identify the slope of the given line:
The equation of the given line is:
[tex]\[
y = -\frac{6}{5} x + b
\][/tex]
Here, the slope ([tex]\(m\)[/tex]) of the line is [tex]\(-\frac{6}{5}\)[/tex].
2. Use the point-slope form of the line equation:
To find the equation of a line parallel to the given line that passes through a specific point, we use the point-slope form of the equation of a line:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope.
3. Substitute the slope and point [tex]\((12, -2)\)[/tex] into the point-slope form:
The point given is [tex]\((12, -2)\)[/tex] and the slope is [tex]\(-\frac{6}{5}\)[/tex]. Substituting these values, we get:
[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 and simplify:
[tex]\[
y + 2 = -\frac{6}{5}x + \frac{6 \cdot 12}{5}
\][/tex]
Simplify further:
[tex]\[
y + 2 = -\frac{6}{5}x + \frac{72}{5}
\][/tex]
5. Solve for [tex]\(y\)[/tex] by isolating the term:
Subtract 2 from both sides:
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - 2
\][/tex]
Convert 2 to a fraction with the same denominator ([tex]\(\frac{10}{5}\)[/tex]):
[tex]\[
y = -\frac{6}{5}x + \frac{72}{5} - \frac{10}{5}
\][/tex]
Combine the fractions:
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
6. Identify the matching equation from the options given:
The equation of the line parallel to [tex]\(y = -\frac{6}{5} x + b\)[/tex] and passing through the point [tex]\((12, -2)\)[/tex] is:
[tex]\[
y = -\frac{6}{5}x + \frac{62}{5}
\][/tex]
Match this with one of the options provided. We look for the option that has the same slope [tex]\(-\frac{6}{5}\)[/tex].
Among the options:
[tex]\[
y = -\frac{6}{5} x + 10
\][/tex]
[tex]\[
y = -\frac{6}{5} x + 12
\][/tex]
[tex]\[
y = -\frac{5}{6} x - 10
\][/tex]
[tex]\[
y = \frac{5}{6} x - 12
\][/tex]
The correct option with slope [tex]\(-\frac{6}{5}\)[/tex] and closest matching is:
[tex]\[
y = -\frac{6}{5} x + 10
\][/tex]
Therefore, the correct answer is option 1: [tex]\( y = -\frac{6}{5} x + 10 \)[/tex].
