Question

Two sidewalks in a park are represented by lines on a coordinate grid. Two points on each of the lines are shown in

the tables.
Sidewalk 1
x y
2 17
0 7
Sidewalk 2
x y
1 20
3 28
(1) Write the equation for Sidewalk 1 in slope-intercept form.
(2) Write the equation for Sidewalk 2 in point-slope form. (You can use either point)
(3) Identify which sidewalk has a steeper slope.

Answer 1

1)
`\sf y = 5x + 7`

2)
`\sf y = 4x + 16`

3) Sidewalk 1 has a steeper slope.

Explanation:

Part (1)

Equation for Sidewalk 1 in Slope-Intercept Form:

To find the equation in slope-intercept form (
`\sf y = mx + b`), where
`\sf m` is the slope and
`\sf b` is the y-intercept, we can use the given points
`\sf (2, 17)` and
`\sf (0, 7)`.

First, find the slope (
`\sf m`):

`\sf m = (y_2 - y_1)/(x_2 - x_1)`

`\sf m = (7 - 17)/(0 - 2)`

`\sf m = (-10)/(-2)`

`\sf m = 5`

Now that we have the slope, choose one of the points (let's use
`\sf (2, 17)`) and substitute into the slope-intercept form equation:

`\sf y = mx + b`

`\sf 17 = 5 \cdot 2 + b`

`\sf 17 = 10 + b`

`\sf b = 7`

So, the equation for Sidewalk 1 is
`\sf y = 5x + 7` in slope-intercept form.

`\hrulefill`

Part (2) Equation for Sidewalk 2 in Point-Slope Form:

To find the equation in point-slope form (
`\sf y - y_1 = m(x - x_1)`), where
`\sf (x_1, y_1)` is a point on the line and
`\sf m` is the slope, we can use the given points
`\sf (1, 20)` and
`\sf (3, 28)`.

First, find the slope (
`\sf m`):

`\sf m = (y_2 - y_1)/(x_2 - x_1)`

`\sf m = (28 - 20)/(3 - 1)`

`\sf m = (8)/(2)`

`\sf m = 4`

Now, choose one of the points (let's use
`\sf (1, 20)`) and substitute into the point-slope form equation:

`\sf y - y_1 = m(x - x_1)`

`\sf y - 20 = 4(x - 1)`

`\sf y - 20 = 4x - 4`

`\sf y = 4x + 16`

So, the equation for Sidewalk 2 is
`\sf y = 4x + 16` in point-slope form.

`\hrulefill`

(3) Identify which sidewalk has a steeper slope:

The slopes of the two sidewalks are as follows:

• Sidewalk 1:
  `\sf m = 5`
• Sidewalk 2:
  `\sf m = 4`

Sidewalk 1 has a steeper slope (5) compared to Sidewalk 2 (4).