To find 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], we follow these steps:
1. Identify the slope of the given line:
The slope-intercept form of a line is given by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
For the line [tex]\( y = -\frac{6}{5}x + 10 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(-\frac{6}{5}\)[/tex].
2. Use the slope for the new line:
Since parallel lines have the same slope, the slope of the new line that we need to find is also [tex]\(-\frac{6}{5}\)[/tex].
3. Apply the point-slope form of the equation:
The point-slope form of a line's equation is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
Here, [tex]\( x_1 = 12 \)[/tex] and [tex]\( y_1 = -2 \)[/tex]. So we substitute these values into the point-slope form:
[tex]\( y - (-2) = -\frac{6}{5}(x - 12) \)[/tex]
Simplify this equation:
[tex]\( y + 2 = -\frac{6}{5}x + \frac{72}{5} \)[/tex]
To convert this into the slope-intercept form [tex]\( y = mx + b \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\( y = -\frac{6}{5}x + \frac{72}{5} - 2 \)[/tex]
4. Convert the constant term:
Convert [tex]\( -2 \)[/tex] to be a fraction with a common denominator of 5:
[tex]\( -2 = -\frac{10}{5} \)[/tex]
Substitute this back into the equation:
[tex]\( y = -\frac{6}{5}x + \frac{72}{5} - \frac{10}{5} \)[/tex]
Combine the constants:
[tex]\( y = -\frac{6}{5}x + \frac{62}{5} \)[/tex]
5. Final equation:
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 equation represents a line parallel to [tex]\( y = -\frac{6}{5}x + 10 \)[/tex] and passes through the point [tex]\( (12, -2) \)[/tex].
