7 Slope Intercept Form Word Problems Solved

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Slope intercept form is a fundamental concept in algebra, and word problems are an excellent way to apply this concept to real-life scenarios. Slope intercept form, also known as the slope-y-intercept form, is a linear equation that represents a straight line on a graph. It's expressed as y = mx + b, where m is the slope of the line, b is the y-intercept, and x is the independent variable.

In this article, we will delve into the world of slope intercept form word problems, providing you with a comprehensive understanding of how to solve them. We'll start by explaining the concept of slope intercept form and then move on to solving seven word problems that will put your algebra skills to the test.

Understanding Slope Intercept Form

Before we dive into the word problems, it's essential to understand the basics of slope intercept form. The equation y = mx + b represents a straight line on a graph, where:

• m is the slope of the line, which represents the rate of change of the dependent variable (y) with respect to the independent variable (x).
• b is the y-intercept, which is the point where the line intersects the y-axis.
• x is the independent variable, which represents the input or the value that we're plugging into the equation.

How to Identify Slope Intercept Form

To identify slope intercept form, look for the following characteristics:

• The equation is in the form y = mx + b.
• The slope (m) is a coefficient of x.
• The y-intercept (b) is a constant term.

Solving Slope Intercept Form Word Problems

Now that we've covered the basics of slope intercept form, let's move on to solving some word problems.

Problem 1: The Cost of Renting a Car

A car rental company charges a base fee of $20 plus an additional $0.25 per mile driven. Write an equation in slope intercept form to represent the total cost of renting a car, where x is the number of miles driven.

Solution:

Let y be the total cost of renting the car. The base fee is $20, and the additional cost is $0.25 per mile driven. Therefore, the equation in slope intercept form is:

y = 0.25x + 20

In this equation, the slope (m) is 0.25, which represents the rate of change of the total cost with respect to the number of miles driven. The y-intercept (b) is 20, which represents the base fee.

Problem 2: The Height of a Tree

A tree grows at a rate of 2 feet per year. If the tree is currently 10 feet tall, write an equation in slope intercept form to represent the height of the tree after x years.

Solution:

Let y be the height of the tree after x years. The tree grows at a rate of 2 feet per year, and the initial height is 10 feet. Therefore, the equation in slope intercept form is:

y = 2x + 10

In this equation, the slope (m) is 2, which represents the rate of change of the height with respect to time. The y-intercept (b) is 10, which represents the initial height of the tree.

Problem 3: The Cost of Electricity

An electricity company charges a base fee of $10 plus an additional $0.05 per kilowatt-hour used. Write an equation in slope intercept form to represent the total cost of electricity, where x is the number of kilowatt-hours used.

Solution:

Let y be the total cost of electricity. The base fee is $10, and the additional cost is $0.05 per kilowatt-hour used. Therefore, the equation in slope intercept form is:

y = 0.05x + 10

In this equation, the slope (m) is 0.05, which represents the rate of change of the total cost with respect to the number of kilowatt-hours used. The y-intercept (b) is 10, which represents the base fee.

Problem 4: The Distance Traveled by a Car

A car travels at a constant speed of 60 miles per hour. If the car travels for x hours, write an equation in slope intercept form to represent the distance traveled.

Solution:

Let y be the distance traveled by the car. The car travels at a constant speed of 60 miles per hour, and the distance traveled is directly proportional to the time traveled. Therefore, the equation in slope intercept form is:

y = 60x

In this equation, the slope (m) is 60, which represents the rate of change of the distance with respect to time. The y-intercept (b) is 0, since the car starts at a distance of 0 miles.

Problem 5: The Height of a Building

A building is 200 feet tall. If a shadow is cast by the building, and the length of the shadow is x feet, write an equation in slope intercept form to represent the height of the building as a function of the length of the shadow.

Solution:

Let y be the height of the building. The length of the shadow is x feet, and the height of the building is directly proportional to the length of the shadow. Since the building is 200 feet tall, the equation in slope intercept form is:

y = (200/x)x

y = 200

In this equation, the slope (m) is 0, since the height of the building is constant. The y-intercept (b) is 200, which represents the height of the building.

Problem 6: The Cost of a Ticket

A ticket to a concert costs $20 plus an additional $5 per person. Write an equation in slope intercept form to represent the total cost of the ticket, where x is the number of people.

Solution:

Let y be the total cost of the ticket. The base fee is $20, and the additional cost is $5 per person. Therefore, the equation in slope intercept form is:

y = 5x + 20

In this equation, the slope (m) is 5, which represents the rate of change of the total cost with respect to the number of people. The y-intercept (b) is 20, which represents the base fee.

Problem 7: The Area of a Rectangle

A rectangle has a length of x feet and a width of 5 feet. Write an equation in slope intercept form to represent the area of the rectangle.

Solution:

Let y be the area of the rectangle. The length of the rectangle is x feet, and the width is 5 feet. Therefore, the equation in slope intercept form is:

y = 5x

In this equation, the slope (m) is 5, which represents the rate of change of the area with respect to the length. The y-intercept (b) is 0, since the area is directly proportional to the length.

Now that we've solved these seven slope intercept form word problems, you should have a better understanding of how to apply this concept to real-life scenarios.