A parks and recreation department is constructing a new bike path.

Question

A parks and recreation department is constructing a new bike path. The path will be parallel to the railroad tracks shown and pass through the parking area at the point (4,5)​ . Write an equation that represents the path.

Answer 1

Hi Madalyn,

You want a path parallel to the railroad tracks, which means you want a path which has the same slope as the railroad tracks. To compute that slope:

m= (y₂-y₁)/(x₂-x₁)

x₁=8

x₂₌₁₁

y₁=0

y₂=4

m= (4-0)/(11-8)

m= 4/3

Now, we want a line that passes through the point (4, 5) with that slope. You didn't specify a form for the equation, so I'll use point-slope. Let me know if you need slope-intercept or general. Anyway, point-slope form is:

y-y₁=m(x-x₁)

x₁=x-coordinate

y₁=y-coordinate

m=slope

So:

x₁=4

y₁=5

m=4/3

Equation: y-5 = (4/3) (x-4)

I hope this helps.

Answer 2

Use the slope formula to find the slope of the line that represents the Main Street.

Points (x₁,y₁)=(8,0) and (x₂,y₂)=(11,4) on the line that represents the Main Street.

Slope (m) = (y₂-y₁) / (x₂-x₁)

= (4-0) / (11-8)

Slope (m) = 4/3

Observed from the figure, a line that passes through the point (4, 5) with that slope.

So use the slope intercept form to find the equation that represents the path as shown below.

(y-y₁) = m (x-x₁)

Substitute(x₁,y₁)=(4,5) and the slope m = 4/3 in the intercept form as shown below.

(y-5) = 4/3 (x-4)

Therefore, the equation that represents the path is y-5 = 4/3 (x-4).