Solve The Equation:${ 4(x + 1) = 2x + 2 }$

Mar 13, 2025 by ADMIN 44 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 linear equation, $4(x + 1) = 2x + 2$ . We will break down the solution into manageable steps, making it easy for readers to understand and follow along.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) 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. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation to be Solved

The equation we will be solving is $4(x + 1) = 2x + 2$ . This equation involves a variable $x$ and constants $4$ , $1$ , $2$ , and $2$ . Our goal is to isolate the variable $x$ and find its value.

Step 1: Distribute the 4

To solve the equation, we need to start by distributing the $4$ to the terms inside the parentheses. This means multiplying the $4$ by each term inside the parentheses.

${ 4(x + 1) = 4x + 4 }$

Step 2: Simplify the Equation

Now that we have distributed the $4$ , we can simplify the equation by combining like terms. In this case, we have $4x$ and $2x$ , which are like terms. We can combine them by adding their coefficients.

${ 4x + 4 = 2x + 2 + 4 }$

Step 3: Combine Like Terms

We can simplify the equation further by combining the constants on the right-hand side.

${ 4x + 4 = 2x + 6 }$

Step 4: Isolate the Variable

Now that we have simplified the equation, we can isolate the variable $x$ by subtracting $2x$ from both sides of the equation.

${ 4x - 2x + 4 = 2x - 2x + 6 }$

Step 5: Simplify the Equation

We can simplify the equation further by combining the like terms.

${ 2x + 4 = 6 }$

Step 6: Solve for x

Finally, we can solve for $x$ by subtracting $4$ from both sides of the equation.

${ 2x = 6 - 4 }$

${ 2x = 2 }$

Step 7: Divide by 2

To find the value of $x$ , we need to divide both sides of the equation by $2$ .

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

${ x = 1 }$

Conclusion

In this article, we solved the linear equation $4(x + 1) = 2x + 2$ using a step-by-step approach. We distributed the $4$ , simplified the equation, combined like terms, isolated the variable, and finally solved for $x$ . The solution to the equation is $x = 1$ . We hope that this article has provided a clear and concise explanation of how to solve linear equations.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

Not distributing the coefficient: Make sure to distribute the coefficient to all terms inside the parentheses.
Not combining like terms: Combine like terms to simplify the equation.
Not isolating the variable: Isolate the variable by adding or subtracting the same value to both sides of the equation.
Not checking the solution: Check the solution by plugging it back into the original equation.

Real-World Applications

Linear equations have many real-world applications. Here are a few examples:

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.

Conclusion

Introduction

In our previous article, we solved the linear equation $4(x + 1) = 2x + 2$ using a step-by-step approach. In this article, we will answer some frequently asked questions about solving linear equations. Whether you're a student struggling with algebra or a professional looking to brush up on your math skills, this Q&A guide is for you.

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 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 solve a linear equation?

A: To solve a linear equation, follow these steps:

Distribute the coefficient: Distribute the coefficient to all terms inside the parentheses.
Combine like terms: Combine like terms to simplify the equation.
Isolate the variable: Isolate the variable by adding or subtracting the same value to both sides of the equation.
Check the solution: Check the solution by plugging it back into the original equation.

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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 4x + 4 = 0$ is a quadratic equation.

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

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your solution by plugging it back into the original equation.

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

A: To determine if an equation is linear or not, look for the highest power of the variable(s). If the highest power is 1, then the equation is linear. If the highest power is greater than 1, then the equation is not linear.

Q: Can I solve linear equations with fractions?

A: Yes, you can solve linear equations with fractions. To do so, follow the same steps as before, but be sure to simplify the fractions as you go.

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

A: The order of operations when solving linear equations is:

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

Q: Can I use algebraic manipulation to solve linear equations?

A: Yes, you can use algebraic manipulation to solve linear equations. Algebraic manipulation involves using properties of equations, such as the commutative and associative properties, to simplify and solve equations.

Conclusion

In conclusion, solving linear equations is an essential skill for students to master. By following the steps outlined in this article and answering the frequently asked questions, readers can become proficient in solving linear equations and apply them to real-world problems. Remember to distribute the coefficient, combine like terms, isolate the variable, and check the solution. With practice and patience, readers can become experts in solving linear equations.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

Not distributing the coefficient: Make sure to distribute the coefficient to all terms inside the parentheses.
Not combining like terms: Combine like terms to simplify the equation.
Not isolating the variable: Isolate the variable by adding or subtracting the same value to both sides of the equation.
Not checking the solution: Check the solution by plugging it back into the original equation.

Real-World Applications

Linear equations have many real-world applications. Here are a few examples:

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.