Question

finding the slope of a line
use the information provided to find the slope of each line. state what the slope represents.
1.

seconds	0	5	10

2.

hours	0	2	5

1. (graph with x-axis hours, y-axis miles, line through origin)
2. (graph with x-axis bags, y-axis pounds, line through origin)
3. (graph with x-axis presses, y-axis ounces, line through origin)
4. (graph with x-axis tickets, y-axis dollars, line through origin)

Explanation:

Let's solve the first problem (the table with Seconds and Feet) as an example:

Step1: Recall the slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

From the table, when \( x_1 = 0 \) (seconds), \( y_1 = 0 \) (feet); when \( x_2 = 5 \) (seconds), \( y_2 = 30 \) (feet).

Step3: Calculate the slope

Substitute into the formula: \( m=\frac{30 - 0}{5 - 0}=\frac{30}{5} = 6 \).

The slope represents the rate of change of distance (feet) with respect to time (seconds), so it's the speed (6 feet per second).

Now for the second problem (Hours and Dollars):

Step1: Recall the slope formula

\( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

From the table, when \( x_1 = 0 \) (hours), \( y_1 = 0 \) (dollars); when \( x_2 = 2 \) (hours), \( y_2 = 18 \) (dollars).

Step3: Calculate the slope

\( m=\frac{18 - 0}{2 - 0}=\frac{18}{2}=9 \).

The slope represents the rate of change of cost (dollars) with respect to time (hours), so it's the hourly rate (9 dollars per hour).

For the third problem (graph with Hours and Miles, assuming a point like (12, 24) from the grid, and (0,0)):

Step1: Recall the slope formula

\( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

Let's take (0, 0) and (12, 24) (from the grid, x=12 hours, y=24 miles).

Step3: Calculate the slope

\( m=\frac{24 - 0}{12 - 0}=\frac{24}{12}=2 \).

The slope represents the speed (2 miles per hour).

For the fourth problem (graph with Bags and Pounds, assuming a point like (10, 4) and (0,0)):

Step1: Recall the slope formula

\( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

Take (0, 0) and (10, 4) (x=10 bags, y=4 pounds).

Step3: Calculate the slope

\( m=\frac{4 - 0}{10 - 0}=\frac{4}{10}=0.4 \).

The slope represents the weight per bag (0.4 pounds per bag).

For the fifth problem (graph with Presses and Ounces, assuming a point like (10, 3) and (0,0)):

Step1: Recall the slope formula

\( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

Take (0, 0) and (10, 3) (x=10 presses, y=3 ounces).

Step3: Calculate the slope

\( m=\frac{3 - 0}{10 - 0}=\frac{3}{10}=0.3 \).

The slope represents the ounces per press (0.3 ounces per press).

For the sixth problem (graph with Tickets and Dollars, assuming a point like (4, 80) and (0,0)):

Step1: Recall the slope formula

\( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

Take (0, 0) and (4, 80) (x=4 tickets, y=80 dollars).

Step3: Calculate the slope

\( m=\frac{80 - 0}{4 - 0}=\frac{80}{4}=20 \).

The slope represents the cost per ticket (20 dollars per ticket).

Answer:

s (for each problem):

1. Slope = 6 (feet per second)
2. Slope = 9 (dollars per hour)
3. Slope = 2 (miles per hour)
4. Slope = 0.4 (pounds per bag)
5. Slope = 0.3 (ounces per press)
6. Slope = 20 (dollars per ticket)