Solve For { X $} . . . { X + \frac{x}{2} = 1 + 2x \}

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which is a simple yet essential equation that requires a clear understanding of algebraic concepts. We will break down the solution step by step, making it easy to follow and understand.

The Equation

The given equation is:

$$x + \frac{x}{2} = 1 + 2x$$

This equation appears to be a simple linear equation, but it requires careful analysis to solve. Our goal is to isolate the variable $x$ and find its value.

Step 1: Simplify the Equation

To simplify the equation, we can start by combining like terms. We can rewrite the equation as:

$$\frac{3x}{2} = 1 + 2x$$

This simplification helps us to focus on the essential terms and makes it easier to solve the equation.

Step 2: Isolate the Variable

To isolate the variable $x$, we need to get rid of the constant term on the right-hand side of the equation. We can do this by subtracting $2x$ from both sides of the equation:

$$\frac{3x}{2} - 2x = 1$$

This step helps us to isolate the variable $x$ and makes it easier to solve the equation.

Step 3: Simplify the Left-Hand Side

Now that we have isolated the variable $x$, we can simplify the left-hand side of the equation. We can rewrite the equation as:

$$\frac{x}{2} = 1$$

This simplification helps us to focus on the essential terms and makes it easier to solve the equation.

Step 4: Solve for $x$

Finally, we can solve for $x$ by multiplying both sides of the equation by 2:

$$x = 2$$

This step gives us the value of the variable $x$, which is the solution to the equation.

Conclusion

Solving linear equations requires a clear understanding of algebraic concepts and a step-by-step approach. By simplifying the equation, isolating the variable, and solving for $x$, we can find the value of the variable. This equation may seem simple, but it requires careful analysis to solve. By following these steps, we can solve linear equations with confidence.

Tips and Tricks

• Always simplify the equation before solving it.
• Isolate the variable by getting rid of the constant term.
• Use inverse operations to solve for the variable.
• Check your solution by plugging it back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects.
• Engineering: Linear equations are used to design and optimize systems.
• Economics: Linear equations are used to model economic systems and make predictions.

Common Mistakes

• Failing to simplify the equation before solving it.
• Not isolating the variable correctly.
• Not checking the solution by plugging it back into the original equation.

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will address some common questions and provide detailed answers to help you better understand how to solve linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

1. Simplify the equation by combining like terms.
2. Isolate the variable by getting rid of the constant term.
3. Use inverse operations to solve for the variable.
4. Check your solution by plugging it back into the original equation.

Q: How do I simplify a linear equation?

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

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

A: To isolate the variable in a linear equation, you need to get rid of the constant term by using inverse operations. For example, if you have the equation x + 2 = 5, you can isolate the variable by subtracting 2 from both sides: x = 3.

Q: What are inverse operations?

A: Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division. When solving a linear equation, you need to use inverse operations to isolate the variable.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug it back into the original equation and see if it is true. For example, if you have the equation x + 2 = 5 and you solve for x, you get x = 3. To check your solution, you can plug x = 3 back into the original equation: 3 + 2 = 5, which is true.

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

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

• Failing to simplify the equation before solving it.
• Not isolating the variable correctly.
• Not checking the solution by plugging it back into the original equation.
• Using the wrong inverse operation.

Q: How do I apply linear equations to real-world problems?

A: Linear equations can be applied to a wide range of real-world problems, including:

• Physics: Linear equations are used to describe the motion of objects.
• Engineering: Linear equations are used to design and optimize systems.
• Economics: Linear equations are used to model economic systems and make predictions.

Q: What are some examples of linear equations in real-world problems?

A: Some examples of linear equations in real-world problems include:

• A car traveling at a constant speed of 60 miles per hour for 2 hours: distance = speed x time = 120 miles.
• A company producing a product at a rate of 100 units per hour for 3 hours: total production = rate x time = 300 units.
• A bank account with a balance of $1000 and a monthly interest rate of 5%: interest = balance x rate = $50.

Conclusion

Solving linear equations is a crucial skill that requires a clear understanding of algebraic concepts and a step-by-step approach. By following these steps and avoiding common mistakes, you can solve linear equations with confidence. Whether you are a student or a professional, mastering linear equations will help you to solve problems and make predictions in various fields.