A railroad construction team wants to calculate an equation to ensure they set a second railroad track parallel to the first. The equation that describes the first railroad track is [tex]y = 3x - 9[/tex]. The second railroad track goes through the point [tex]\((2, 9)\)[/tex].
What is the equation, in slope-intercept form, of the second railroad track?
To find the equation of the second railroad track in slope-intercept form, we need to determine both the slope and the y-intercept of the line.
1. Identify the slope of the first railroad track: The equation of the first railroad track is [tex]\( y = 3x - 9 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is 3.
2. Given point on the second railroad track: The second railroad track passes through the point [tex]\( (2, 9) \)[/tex].
3. Determine the slope of the second railroad track: Since the second railroad track is parallel to the first, it has the same slope. Thus, the slope [tex]\( m \)[/tex] of the second railroad track is also 3.
4. Use the point-slope formula to determine the y-intercept [tex]\( b \)[/tex]: The point-slope form of a line's equation is [tex]\( y = mx + b \)[/tex]. Substitute the given point [tex]\( (2, 9) \)[/tex] and the slope [tex]\( m = 3 \)[/tex] into the equation:
[tex]\[
y = mx + b
\][/tex]
Substitute [tex]\( x = 2 \)[/tex], [tex]\( y = 9 \)[/tex], and [tex]\( m = 3 \)[/tex]:
[tex]\[
9 = 3(2) + b
\][/tex]
Simplify to solve for [tex]\( b \)[/tex]:
[tex]\[
9 = 6 + b \implies b = 3
\][/tex]
5. Write the equation in slope-intercept form: Now that we have the slope [tex]\( m = 3 \)[/tex] and the y-intercept [tex]\( b = 3 \)[/tex], the equation of the second railroad track is:
[tex]\[
y = 3x + 3
\][/tex]
So, the correct equation in slope-intercept form of the second railroad track is: [tex]\[ y = 3x + 3 \][/tex]
