Solve For $u$.$3u + 9 - 11u = -23$Simplify Your Answer As Much As Possible.

Feb 27, 2025 by ADMIN 80 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, where the variable is isolated on one side of the equation. We will use the given equation $3u + 9 - 11u = -23$ as an example to demonstrate the step-by-step process of solving for $u$.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, $u$) is 1. The equation is also a simple linear equation, as it does not involve any quadratic or higher-order terms.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by combining like terms. In this case, we have two terms involving $u$, namely $3u$ and $-11u$. We can combine these terms by adding their coefficients, which gives us $-8u$.

# Simplifying the equation
equation = "3u + 9 - 11u = -23"
simplified_equation = "-8u + 9 = -23"
print(simplified_equation)

Step 2: Isolate the Variable

Now that we have simplified the equation, we need to isolate the variable $u$. To do this, we can add $11u$ to both sides of the equation, which will eliminate the $-11u$ term. This gives us:

3u + 9 = -23 + 11u

# Isolating the variable
simplified_equation = "-8u + 9 = -23"
isolated_equation = "3u + 9 = -23 + 11u"
print(isolated_equation)

Step 3: Solve for $u$

Now that we have isolated the variable $u$, we can solve for it by subtracting $3u$ from both sides of the equation. This gives us:

9 = -23 + 14u

# Solving for u
isolated_equation = "3u + 9 = -23 + 11u"
solved_equation = "9 = -23 + 14u"
print(solved_equation)

Step 4: Simplify the Equation

The final step is to simplify the equation by combining like terms. In this case, we have a constant term $-23$ on the right-hand side of the equation. We can simplify the equation by adding $23$ to both sides, which gives us:

32 = 14u

# Simplifying the equation
solved_equation = "9 = -23 + 14u"
simplified_equation = "32 = 14u"
print(simplified_equation)

Step 5: Solve for $u$

Finally, we can solve for $u$ by dividing both sides of the equation by $14$. This gives us:

u = \frac{32}{14}

# Solving for u
simplified_equation = "32 = 14u"
solution = "u = 32/14"
print(solution)

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation. We started with the given equation $3u + 9 - 11u = -23$ and simplified it by combining like terms. We then isolated the variable $u$ by adding $11u$ to both sides of the equation. Finally, we solved for $u$ by dividing both sides of the equation by $14$. The solution to the equation is $u = \frac{32}{14}$.

Final Answer

The final answer is $\boxed{\frac{16}{7}}$.

Additional Resources

For more information on solving linear equations, please refer to the following resources:

Khan Academy: Solving Linear Equations
Mathway: Solving Linear Equations
Wolfram Alpha: Solving Linear Equations

FAQs

Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable is 1.

Q: How do I simplify a linear equation? A: To simplify a linear equation, combine like terms by adding or subtracting the coefficients of the variable.

Q: How do I isolate the variable in a linear equation? A: To isolate the variable, add or subtract the same value to both sides of the equation.

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will provide a comprehensive Q&A guide on solving linear equations. Whether you are a student, teacher, or simply looking to brush up on your math skills, this guide will provide you with the answers to your most frequently asked questions.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, combine like terms by adding or subtracting the coefficients of the variable. For example, if you have the equation 2x + 3x = 5, you can simplify it by combining the like terms: 5x = 5.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable, add or subtract the same value to both sides of the equation. For example, if you have the equation 2x + 3 = 5, you can isolate the variable by subtracting 3 from both sides: 2x = 2.

Q: How do I solve for the variable in a linear equation?

A: To solve for the variable, divide both sides of the equation by the coefficient of the variable. For example, if you have the equation 2x = 4, you can solve for x by dividing both sides by 2: x = 2.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 2x + 1 = 0 is a quadratic equation.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the method of substitution or elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the importance of solving linear equations?

A: Solving linear equations is an essential skill in mathematics and is used in a wide range of applications, including science, engineering, economics, and finance. It is also a fundamental concept in algebra and is used to solve more complex equations.

Q: How do I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

Using online resources, such as Khan Academy or Mathway
Working with a tutor or teacher
Practicing with worksheets or exercises
Solving real-world problems that involve linear equations

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Not simplifying the equation before solving it
Not isolating the variable correctly
Not checking the solution to make sure it is correct
Not using the correct method to solve the equation

Conclusion

Solving linear equations is a fundamental concept in mathematics that is used in a wide range of applications. By understanding the basics of linear equations and practicing solving them, you can develop a strong foundation in algebra and math. Remember to simplify the equation, isolate the variable, and solve for the variable correctly to ensure that you get the right answer.