To determine the equation of the second railroad track, we need it to be parallel to the first track and pass through the given point [tex]\((2, 9)\)[/tex].
The equation of the first track is [tex]\(y = 3x - 9\)[/tex].
1. Identify the slope of the first track:
The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. From the equation [tex]\(y = 3x - 9\)[/tex], we can see that the slope ([tex]\(m\)[/tex]) is [tex]\(3\)[/tex].
2. Use the slope for the second track:
Parallel lines have the same slope. Therefore, the slope of the second track is also [tex]\(3\)[/tex].
3. Find the y-intercept of the second track:
We use the point-slope form of the equation, which is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where:
- [tex]\(m\)[/tex] is the slope,
- [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes.
Here, the point is [tex]\((2, 9)\)[/tex], and the slope [tex]\(m\)[/tex] is [tex]\(3\)[/tex]. Substituting these values into the point-slope form:
[tex]\[
y - 9 = 3(x - 2)
\][/tex]
4. Simplify this equation:
Expand and rearrange the equation to get it into the slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[
y - 9 = 3x - 6
\][/tex]
[tex]\[
y = 3x - 6 + 9
\][/tex]
[tex]\[
y = 3x + 3
\][/tex]
So, the equation of the second railroad track in slope-intercept form is:
[tex]\[ y = 3x + 3 \][/tex]
