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
feet: 0, 30, 60
2
hours: 0, 2, 5
dollars: 0, 18, 45
3
graph with x-axis (hours) and y-axis (miles), line from (0,0)
4
graph with x-axis (bags) and y-axis (pounds), line from (0,0)

Explanation:

Step1: Recall slope formula

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Solve Problem 1

For the first table, we have points \((0,0)\), \((5,30)\), \((10,60)\). Take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(5,30)\). Then \( m=\frac{30 - 0}{5 - 0}=\frac{30}{5} = 6 \). The slope represents the speed (feet per second), as it is the rate of change of distance (feet) with respect to time (seconds).

Step3: Solve Problem 2

For the second table, points are \((0,0)\), \((2,18)\), \((5,45)\). Take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(2,18)\). Then \( m=\frac{18 - 0}{2 - 0}=\frac{18}{2}=9 \). The slope represents the rate of pay (dollars per hour), the rate of change of money (dollars) with respect to time (hours).

Step4: Solve Problem 3

From the graph, we can take two points. Let's take \((0,0)\) and \((1,2)\) (by looking at the grid, when \( x = 1 \) (hours), \( y=2 \) (miles)). Then \( m=\frac{2 - 0}{1 - 0}=2 \). The slope represents the speed (miles per hour), the rate of change of distance (miles) with respect to time (hours).

Step5: Solve Problem 4

From the graph, take two points. Let's take \((0,0)\) and \((2,1)\) (when \( x = 2 \) (bags), \( y = 1 \) (pounds)). Then \( m=\frac{1 - 0}{2 - 0}=\frac{1}{2}=0.5 \). The slope represents the weight per bag (pounds per bag), the rate of change of weight (pounds) with respect to the number of bags.

Answer:

1. Slope = 6 (represents speed: 6 feet per second)
2. Slope = 9 (represents rate of pay: 9 dollars per hour)
3. Slope = 2 (represents speed: 2 miles per hour)
4. Slope = 0.5 (represents weight per bag: 0.5 pounds per bag)