To find the equation of a line parallel to a given line and passing through a specific point, follow these steps:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = 6x + 7 \)[/tex]. The slope-intercept form of a line's equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
From the equation [tex]\( y = 6x + 7 \)[/tex], we see that the slope [tex]\( m \)[/tex] is 6.
2. Determine the slope of the parallel line:
Lines that are parallel to each other have the same slope. Therefore, the slope of the line we need to find is also 6.
3. Use the point-slope form of the equation of the line:
The point-slope form 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.
We are given the point [tex]\( (0, 9) \)[/tex]. Let's substitute [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = 9 \)[/tex], and [tex]\( m = 6 \)[/tex] into the point-slope form:
[tex]\[
y - 9 = 6(x - 0)
\][/tex]
4. Simplify the equation:
Simplify the above equation to get the slope-intercept form:
[tex]\[
y - 9 = 6x
\][/tex]
[tex]\[
y = 6x + 9
\][/tex]
So, the equation of the line that is parallel to [tex]\( y = 6x + 7 \)[/tex] and passes through the point [tex]\( (0, 9) \)[/tex] is:
[tex]\[
y = 6x + 9
\][/tex]
