Question

the city is planning to build a new bike path. the path will run parallel to the main road, which can be modeled by the equation y = 2x+5. the bike path will pass through a community park located at the coordinates (3,1). 20) write the equation in slope - intercept form for the line representing the new bike path. 21) a new rest stop is to be built on a line perpendicular to the main road, also passing through the park at (3,1). write the equation of this line. two lanes of a race track run parallel to each other. the outer lane is represented by the equation y = 0.5x−2. the inner lane passes through the point (4,0). 22) what is the slope of the inner lane? explain your reasoning. 23) write the equation in slope - intercept form for the inner lane. 24) a safety barrier is to be placed at a right angle to the inner lane at the point (4,0). write the equation of the line representing this safety barrier.

Explanation:

Step1: Recall slope - parallel lines

Parallel lines have the same slope. The equation of the main road is $y = 2x+5$, so the slope of the bike - path (parallel to the main road) is $m = 2$.

Step2: Use point - slope form to find y - intercept

The point - slope form is $y - y_1=m(x - x_1)$. Substitute $m = 2$, $x_1 = 3$, and $y_1 = 1$: $y-1=2(x - 3)$. Expand to get $y-1=2x-6$. Then solve for $y$: $y=2x-5$.

Step1: Recall slope - perpendicular lines

The slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$. The slope of the main road is $m = 2$, so the slope of the perpendicular line is $m=-\frac{1}{2}$.

Step2: Use point - slope form to find equation

Using the point - slope form $y - y_1=m(x - x_1)$ with $m =-\frac{1}{2}$, $x_1 = 3$, and $y_1 = 1$. We have $y - 1=-\frac{1}{2}(x - 3)$. Expand: $y-1=-\frac{1}{2}x+\frac{3}{2}$. Then solve for $y$: $y=-\frac{1}{2}x+\frac{5}{2}$.

Step1: Recall slope of parallel lines

Parallel lines have the same slope. The outer lane is $y = 0.5x-2$, so the slope of the inner lane (parallel to the outer lane) is $m = 0.5$ because parallel lines have equal slopes.

Step1: Use point - slope form with known slope and point

The slope of the inner lane is $m = 0.5$ and it passes through $(4,0)$. Using the point - slope form $y - y_1=m(x - x_1)$, we have $y-0=0.5(x - 4)$. Expand: $y=0.5x - 2$.

Answer:

$y = 2x-5$