A Van's Daily Profit Can Be Represented By The Equation $y = 5x + 15$, Where $x$ Represents The Number Of Hours Worked And $y$ Represents The Profit In Dollars.Find Both The $x$-intercept And $y$-intercept

Introduction

In the world of business, understanding the relationship between the number of hours worked and the profit earned is crucial for making informed decisions. A van's daily profit can be represented by the equation $y = 5x + 15$, where $x$ represents the number of hours worked and $y$ represents the profit in dollars. In this article, we will delve into the world of linear equations and find both the $x$-intercept and $y$-intercept of the given equation.

What are Intercepts?

In the context of linear equations, intercepts refer to the points where the graph of the equation intersects the $x$-axis and the $y$-axis. The $x$-intercept is the point where the graph intersects the $x$-axis, and the $y$-intercept is the point where the graph intersects the $y$-axis.

Finding the $x$-Intercept

To find the $x$-intercept, we need to set $y$ equal to zero and solve for $x$. This is because the $x$-intercept occurs when the profit is zero, i.e., $y = 0$. Substituting $y = 0$ into the equation $y = 5x + 15$, we get:

$$0 = 5x + 15$$

Subtracting 15 from both sides of the equation, we get:

$$-15 = 5x$$

Dividing both sides of the equation by 5, we get:

$$x = -3$$

Therefore, the $x$-intercept is $(-3, 0)$.

Finding the $y$-Intercept

To find the $y$-intercept, we need to set $x$ equal to zero and solve for $y$. This is because the $y$-intercept occurs when the number of hours worked is zero, i.e., $x = 0$. Substituting $x = 0$ into the equation $y = 5x + 15$, we get:

$$y = 5(0) + 15$$

Simplifying the equation, we get:

$$y = 15$$

Therefore, the $y$-intercept is $(0, 15)$.

Conclusion

In conclusion, we have found both the $x$-intercept and the $y$-intercept of the given equation $y = 5x + 15$. The $x$-intercept is $(-3, 0)$, and the $y$-intercept is $(0, 15)$. Understanding the intercepts of a linear equation is crucial in various fields, including business, economics, and engineering.

Real-World Applications

The concept of intercepts has numerous real-world applications. For instance, in business, understanding the relationship between the number of hours worked and the profit earned can help entrepreneurs make informed decisions about pricing, production, and marketing. In economics, intercepts can be used to analyze the relationship between variables such as GDP, inflation, and unemployment. In engineering, intercepts can be used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.

Example Problems

Here are a few example problems to help reinforce the concept of intercepts:

• Find the $x$-intercept and the $y$-intercept of the equation $y = 2x + 10$.
• Find the $x$-intercept and the $y$-intercept of the equation $y = 3x - 5$.
• Find the $x$-intercept and the $y$-intercept of the equation $y = 4x + 20$.

Solutions

Here are the solutions to the example problems:

• For the equation $y = 2x + 10$, the $x$-intercept is $(-5, 0)$, and the $y$-intercept is $(0, 10)$.
• For the equation $y = 3x - 5$, the $x$-intercept is $(-\frac{5}{3}, 0)$, and the $y$-intercept is $(0, -5)$.
• For the equation $y = 4x + 20$, the $x$-intercept is $(-5, 0)$, and the $y$-intercept is $(0, 20)$.

Conclusion

$$$$

Q: What is the equation for a van's daily profit?

A: The equation for a van's daily profit is $y = 5x + 15$, where $x$ represents the number of hours worked and $y$ represents the profit in dollars.

Q: What is the $x$-intercept of the equation?

A: The $x$-intercept of the equation is $(-3, 0)$, which means that when the van works for 3 hours, the profit is zero.

Q: What is the $y$-intercept of the equation?

A: The $y$-intercept of the equation is $(0, 15)$, which means that when the van does not work, the profit is $15.

Q: How do I find the $x$-intercept of a linear equation?

A: To find the $x$-intercept of a linear equation, set $y$ equal to zero and solve for $x$. This is because the $x$-intercept occurs when the profit is zero.

Q: How do I find the $y$-intercept of a linear equation?

A: To find the $y$-intercept of a linear equation, set $x$ equal to zero and solve for $y$. This is because the $y$-intercept occurs when the number of hours worked is zero.

Q: What are some real-world applications of intercepts?

A: Intercepts have numerous real-world applications, including business, economics, and engineering. For example, in business, understanding the relationship between the number of hours worked and the profit earned can help entrepreneurs make informed decisions about pricing, production, and marketing.

Q: Can you provide some example problems to help reinforce the concept of intercepts?

A: Here are a few example problems:

• Find the $x$-intercept and the $y$-intercept of the equation $y = 2x + 10$.
• Find the $x$-intercept and the $y$-intercept of the equation $y = 3x - 5$.
• Find the $x$-intercept and the $y$-intercept of the equation $y = 4x + 20$.

Q: Can you provide the solutions to the example problems?

A: Here are the solutions to the example problems:

• For the equation $y = 2x + 10$, the $x$-intercept is $(-5, 0)$, and the $y$-intercept is $(0, 10)$.
• For the equation $y = 3x - 5$, the $x$-intercept is $(-\frac{5}{3}, 0)$, and the $y$-intercept is $(0, -5)$.
• For the equation $y = 4x + 20$, the $x$-intercept is $(-5, 0)$, and the $y$-intercept is $(0, 20)$.

Q: How can I use intercepts in real-world applications?

A: Intercepts can be used in various real-world applications, including:

• Business: Understanding the relationship between the number of hours worked and the profit earned can help entrepreneurs make informed decisions about pricing, production, and marketing.
• Economics: Intercepts can be used to analyze the relationship between variables such as GDP, inflation, and unemployment.
• Engineering: Intercepts can be used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.

Q: What are some common mistakes to avoid when working with intercepts?

A: Some common mistakes to avoid when working with intercepts include:

• Failing to set $y$ equal to zero to find the $x$-intercept.
• Failing to set $x$ equal to zero to find the $y$-intercept.
• Not checking the signs of the coefficients in the equation.
• Not considering the domain and range of the function.