The slope-intercept form is one of the most widely used forms of linear equations, and it's essential to understand how to apply it to real-life situations. In this article, we'll explore five real-life word problems that can be solved using the slope-intercept form.
What is Slope-Intercept Form?
Before diving into the word problems, let's quickly review the slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The slope represents the rate of change of the line, while the y-intercept is the point where the line crosses the y-axis.
Word Problem 1: Cost of Renting a Bike
A bike rental company charges a base fee of $10 plus an additional $5 per hour. If you rent a bike for 4 hours, how much will you pay in total? We can use the slope-intercept form to solve this problem.
Let's say the total cost is y and the number of hours is x. We can write the equation as y = 5x + 10. To find the total cost for 4 hours, we can plug in x = 4 into the equation:
y = 5(4) + 10 y = 20 + 10 y = 30
Therefore, you will pay a total of $30 to rent a bike for 4 hours.
How Does the Slope-Intercept Form Help?
In this problem, the slope-intercept form helps us understand the relationship between the cost and the number of hours. The slope (5) represents the rate of change, which is the additional cost per hour. The y-intercept (10) represents the base fee.
Word Problem 2: Distance and Time
A car travels from City A to City B at a constant speed. If it takes 3 hours to travel 120 miles, how far will it travel in 5 hours? We can use the slope-intercept form to solve this problem.
Let's say the distance traveled is y and the time is x. We can write the equation as y = 40x. To find the distance traveled in 5 hours, we can plug in x = 5 into the equation:
y = 40(5) y = 200
Therefore, the car will travel 200 miles in 5 hours.
How Does the Slope-Intercept Form Help?
In this problem, the slope-intercept form helps us understand the relationship between distance and time. The slope (40) represents the speed of the car, which is the rate of change. The y-intercept is 0, which means the car starts at a distance of 0 miles.
Word Problem 3: Profit and Sales
A company sells a product for $20 each. If the company's fixed costs are $100 and the variable costs are $5 per unit, how many units must the company sell to make a profit of $500? We can use the slope-intercept form to solve this problem.
Let's say the profit is y and the number of units sold is x. We can write the equation as y = 15x - 100. To find the number of units sold to make a profit of $500, we can plug in y = 500 into the equation:
500 = 15x - 100 600 = 15x x = 40
Therefore, the company must sell 40 units to make a profit of $500.
How Does the Slope-Intercept Form Help?
In this problem, the slope-intercept form helps us understand the relationship between profit and sales. The slope (15) represents the profit per unit, which is the rate of change. The y-intercept (-100) represents the fixed costs.
Word Problem 4: Height and Age
A child's height is modeled by the equation y = 2x + 30, where y is the height in inches and x is the age in years. If the child is 10 years old, how tall is he? We can use the slope-intercept form to solve this problem.
We can plug in x = 10 into the equation:
y = 2(10) + 30 y = 20 + 30 y = 50
Therefore, the child is 50 inches tall.
How Does the Slope-Intercept Form Help?
In this problem, the slope-intercept form helps us understand the relationship between height and age. The slope (2) represents the rate of growth, which is the rate of change. The y-intercept (30) represents the initial height.
Word Problem 5: Electricity Bill
An electricity company charges a base fee of $20 plus an additional $0.05 per kilowatt-hour (kWh). If a household uses 1000 kWh of electricity, how much will the electricity bill be? We can use the slope-intercept form to solve this problem.
Let's say the electricity bill is y and the number of kWh is x. We can write the equation as y = 0.05x + 20. To find the electricity bill for 1000 kWh, we can plug in x = 1000 into the equation:
y = 0.05(1000) + 20 y = 50 + 20 y = 70
Therefore, the electricity bill will be $70.
How Does the Slope-Intercept Form Help?
In this problem, the slope-intercept form helps us understand the relationship between the electricity bill and the number of kWh. The slope (0.05) represents the rate of change, which is the additional cost per kWh. The y-intercept (20) represents the base fee.
Conclusion
The slope-intercept form is a powerful tool for solving real-life word problems. By understanding the relationship between the variables and the rate of change, we can make informed decisions and predictions. Whether it's calculating the cost of renting a bike, determining the distance traveled by a car, or predicting the profit of a company, the slope-intercept form is an essential skill to have.
We hope this article has helped you understand the importance of the slope-intercept form in real-life applications. If you have any questions or comments, please don't hesitate to share them below.
What is the slope-intercept form?
+The slope-intercept form is a linear equation in the form y = mx + b, where m is the slope and b is the y-intercept.
How do I use the slope-intercept form to solve word problems?
+To use the slope-intercept form to solve word problems, identify the variables and the rate of change, and then plug in the values into the equation.
What are some common applications of the slope-intercept form?
+The slope-intercept form has many real-life applications, including calculating the cost of goods, determining the distance traveled by a car, and predicting the profit of a company.