Solving for y in the Equation

Oct 13, 2025 by magicalceylontours.com 106 views

Introduction

In mathematics, equations are a fundamental concept that helps us understand and describe various relationships between variables. Solving equations is a crucial skill that is used in various fields, including physics, engineering, economics, and more. In this article, we will focus on solving a specific equation and finding the restriction on the domain.

Solving for y in the Equation

The given equation is:

${ \frac{x}{1} = \frac{-2y + 5}{3y + 1} }$

To solve for y, we can start by cross-multiplying both sides of the equation:

${ x(3y + 1) = -2y + 5 }$

Expanding the left side of the equation, we get:

${ 3xy + x = -2y + 5 }$

Now, let's move all the terms involving y to one side of the equation and the constant terms to the other side:

${ 3xy + 2y = 5 - x }$

Factoring out y from the left side of the equation, we get:

${ y(3x + 2) = 5 - x }$

Now, we can solve for y by dividing both sides of the equation by (3x + 2):

${ y = \frac{5 - x}{3x + 2} }$

Finding the Restriction on the Domain

The domain of a function is the set of all possible input values for which the function is defined. In this case, we need to find the restriction on the domain of the function y = (5 - x) / (3x + 2).

To find the restriction on the domain, we need to identify the values of x that would make the denominator (3x + 2) equal to zero. This is because division by zero is undefined.

Setting the denominator equal to zero, we get:

${ 3x + 2 = 0 }$

Solving for x, we get:

${ 3x = -2 }$

${ x = -\frac{2}{3} }$

Therefore, the restriction on the domain is x ≠ -2/3.

Conclusion

In this article, we solved the equation y = (5 - x) / (3x + 2) and found the restriction on the domain. The solution to the equation is y = (5 - x) / (3x + 2), and the restriction on the domain is x ≠ -2/3. This restriction ensures that the function is defined for all values of x except x = -2/3.

Example Use Cases

Physics: In physics, equations are used to describe the motion of objects. For example, the equation y = (5 - x) / (3x + 2) could be used to describe the position of an object as a function of time.
Engineering: In engineering, equations are used to design and optimize systems. For example, the equation y = (5 - x) / (3x + 2) could be used to design a control system for a robot.
Economics: In economics, equations are used to model economic systems. For example, the equation y = (5 - x) / (3x + 2) could be used to model the demand for a product.

Tips and Tricks

Simplifying Equations: When solving equations, it's often helpful to simplify the equation by combining like terms or canceling out common factors.
Checking Solutions: When solving equations, it's always a good idea to check your solutions by plugging them back into the original equation.
Graphing Functions: Graphing functions can be a helpful way to visualize the behavior of a function and identify any restrictions on the domain.

Common Mistakes

Dividing by Zero: One common mistake when solving equations is to divide by zero. This is because division by zero is undefined.
Not Checking Solutions: Another common mistake is to not check solutions by plugging them back into the original equation.
Not Simplifying Equations: Failing to simplify equations can make it difficult to solve the equation and can lead to incorrect solutions.

Conclusion

Introduction

In our previous article, we solved the equation y = (5 - x) / (3x + 2) and found the restriction on the domain. In this article, we will answer some frequently asked questions about solving equations and finding domain restrictions.

Q: What is the difference between a function and an equation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). An equation, on the other hand, is a statement that two expressions are equal. In other words, a function is a way of describing a relationship between variables, while an equation is a specific instance of that relationship.

Q: How do I know if an equation is a function or not?

A: To determine if an equation is a function, you need to check if it passes the vertical line test. If a vertical line intersects the graph of the equation at more than one point, then the equation is not a function. If a vertical line intersects the graph of the equation at only one point, then the equation is a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is defined.

Q: How do I find the domain of a function?

A: To find the domain of a function, you need to identify any values of x that would make the denominator of the function equal to zero. This is because division by zero is undefined. You also need to check if there are any other restrictions on the domain, such as values of x that would make the function undefined.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of y for which the function is defined.

Q: How do I find the range of a function?

A: To find the range of a function, you need to identify any values of y that would make the function undefined. You also need to check if there are any other restrictions on the range, such as values of y that would make the function undefined.

Q: What is the difference between a linear function and a nonlinear function?

A: A linear function is a function that can be written in the form y = mx + b, where m and b are constants. A nonlinear function, on the other hand, is a function that cannot be written in this form. Nonlinear functions can have a wide range of shapes and behaviors.

Q: How do I graph a function?

A: To graph a function, you need to identify the x and y intercepts of the function. You also need to identify any other key features of the function, such as its maximum and minimum values. You can use a graphing calculator or a computer program to graph the function.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A relation, on the other hand, is a set of ordered pairs that satisfy a certain condition. In other words, a function is a way of describing a relationship between variables, while a relation is a set of specific instances of that relationship.

Conclusion

In conclusion, solving equations and finding domain restrictions are essential skills in mathematics. By following the steps outlined in this article, you can solve equations and find domain restrictions with confidence. Remember to simplify equations, check solutions, and avoid common mistakes like dividing by zero. With practice and patience, you can become proficient in solving equations and finding domain restrictions.

Example Use Cases

Physics: In physics, equations are used to describe the motion of objects. For example, the equation y = (5 - x) / (3x + 2) could be used to describe the position of an object as a function of time.
Engineering: In engineering, equations are used to design and optimize systems. For example, the equation y = (5 - x) / (3x + 2) could be used to design a control system for a robot.
Economics: In economics, equations are used to model economic systems. For example, the equation y = (5 - x) / (3x + 2) could be used to model the demand for a product.

Tips and Tricks

Simplifying Equations: When solving equations, it's often helpful to simplify the equation by combining like terms or canceling out common factors.
Checking Solutions: When solving equations, it's always a good idea to check your solutions by plugging them back into the original equation.
Graphing Functions: Graphing functions can be a helpful way to visualize the behavior of a function and identify any restrictions on the domain.

Common Mistakes

Dividing by Zero: One common mistake when solving equations is to divide by zero. This is because division by zero is undefined.
Not Checking Solutions: Another common mistake is to not check solutions by plugging them back into the original equation.
Not Simplifying Equations: Failing to simplify equations can make it difficult to solve the equation and can lead to incorrect solutions.