Solve The Equation For { X $}$: ${ 4 = 3(2x + 1) - 11 }$

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 linear equation, step by step, to help readers understand the process and build their confidence in solving similar equations.

The Equation

The equation we will be solving is:

$$4 = 3(2x + 1) - 11$$

This equation appears to be complex, but with a systematic approach, we can break it down and solve for the variable $x$.

Step 1: Distribute the Coefficient

The first step in solving this equation is to distribute the coefficient 3 to the terms inside the parentheses.

$$4 = 3(2x + 1) - 11$$

$$4 = 6x + 3 - 11$$

Step 2: Combine Like Terms

Now that we have distributed the coefficient, we can combine like terms on the right-hand side of the equation.

$$4 = 6x + 3 - 11$$

$$4 = 6x - 8$$

Step 3: Add or Subtract the Coefficient

The next step is to isolate the variable $x$ by adding or subtracting the coefficient to both sides of the equation.

$$4 = 6x - 8$$

$$4 + 8 = 6x - 8 + 8$$

$$12 = 6x$$

Step 4: Divide Both Sides

Finally, we can solve for $x$ by dividing both sides of the equation by the coefficient 6.

$$12 = 6x$$

$$\frac{12}{6} = \frac{6x}{6}$$

$$2 = x$$

Conclusion

In this article, we have solved a linear equation step by step, using a systematic approach to isolate the variable $x$. By following these steps, readers can build their confidence in solving similar equations and develop a deeper understanding of linear equations.

Tips and Tricks

• Always start by distributing the coefficient to the terms inside the parentheses.
• Combine like terms on the right-hand side of the equation.
• Add or subtract the coefficient to both sides of the equation to isolate the variable.
• Divide both sides of the equation by the coefficient to solve for the variable.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

• Failing to distribute the coefficient to the terms inside the parentheses.
• Not combining like terms on the right-hand side of the equation.
• Not adding or subtracting the coefficient to both sides of the equation.
• Not dividing both sides of the equation by the coefficient.

Practice Problems

Try solving the following linear equations:

1. $2x + 5 = 11$
2. $3x - 2 = 7$
3. $4x + 2 = 12$

Solutions

1. $2x + 5 = 11$

$$2x + 5 - 5 = 11 - 5$$

$$2x = 6$$

$$\frac{2x}{2} = \frac{6}{2}$$

$$x = 3$$

2. $3x - 2 = 7$

$$3x - 2 + 2 = 7 + 2$$

$$3x = 9$$

$$\frac{3x}{3} = \frac{9}{3}$$

$$x = 3$$

3. $4x + 2 = 12$

$$4x + 2 - 2 = 12 - 2$$

$$4x = 10$$

$$\frac{4x}{4} = \frac{10}{4}$$

x = \frac{5}{2}$<br/> **Solving Linear Equations: A Q&A Guide** =====================================

Introduction

In our previous article, we explored the step-by-step process of solving a linear equation. However, we understand that sometimes, it's not just about following a set of steps, but also about understanding the underlying concepts and addressing common questions and concerns. In this article, we'll delve into a Q&A format, addressing some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it's 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: What is the difference between a linear equation and a quadratic equation?

A: A linear equation has the highest power of the variable as 1, whereas a quadratic equation has the highest power of the variable as 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, look for the highest power of the variable. If it's 1, it's a linear equation. If it's 2, it's a quadratic equation.

Q: What is the order of operations when solving a linear equation?

A: When solving a linear equation, follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: To isolate the variable in a linear equation, follow these steps:

1. Add or subtract the same value to both sides of the equation to get rid of any constants on the same side as the variable.
2. Multiply or divide both sides of the equation by the same value to get rid of any coefficients on the same side as the variable.

Q: What is the difference between a coefficient and a constant?

A: A coefficient is a number that multiplies a variable, while a constant is a number that doesn't change value. For example, in the equation 2x + 3 = 5, 2 is the coefficient of x, and 3 is the constant.

Q: How do I handle negative coefficients in a linear equation?

A: When dealing with negative coefficients, simply multiply both sides of the equation by -1 to get rid of the negative sign.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, make sure to follow the order of operations and check your work to ensure accuracy.

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 distribute the coefficient to the terms inside the parentheses.
• Not combining like terms on the same side of the equation.
• Not adding or subtracting the same value to both sides of the equation.
• Not multiplying or dividing both sides of the equation by the same value.

Conclusion

Solving linear equations can seem daunting at first, but with practice and patience, you'll become more confident and proficient. Remember to follow the order of operations, isolate the variable, and avoid common mistakes. If you have any further questions or concerns, feel free to ask!