Solve The Equation: $ 2(2x + 1) = 3(3 - X) $

Mar 1, 2025 by ADMIN 45 views

**Solving Linear Equations: A Step-by-Step Guide to $2(2x + 1) = 3(3 - x)$**

Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a specific linear equation, $2(2x + 1) = 3(3 - x)$ , using step-by-step instructions.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand what it represents. The equation $2(2x + 1) = 3(3 - x)$ is a linear equation that involves two variables, $x$ and constants. The equation is a statement that two expressions are equal, and our goal is to find the value of $x$ that makes the equation true.

Step 1: Distribute the Numbers Outside the Parentheses

To solve the equation, we need to start by distributing the numbers outside the parentheses to the terms inside. This will help us simplify the equation and make it easier to work with.

2(2x + 1) = 2(2x) + 2(1)
= 4x + 2

3(3 - x) = 3(3) - 3(x)
= 9 - 3x

Step 2: Simplify the Equation

Now that we have distributed the numbers outside the parentheses, we can simplify the equation by combining like terms.

4x + 2 = 9 - 3x

Step 3: Add 3x to Both Sides of the Equation

To isolate the variable $x$ , we need to get all the terms with $x$ on one side of the equation. We can do this by adding $3x$ to both sides of the equation.

4x + 2 + 3x = 9 - 3x + 3x
= 4x + 3x + 2 = 9
= 7x + 2 = 9

Step 4: Subtract 2 from Both Sides of the Equation

Next, we need to get rid of the constant term on the left side of the equation. We can do this by subtracting 2 from both sides of the equation.

7x + 2 - 2 = 9 - 2
= 7x = 7

Step 5: Divide Both Sides of the Equation by 7

Finally, we need to solve for $x$ by dividing both sides of the equation by 7.

7x / 7 = 7 / 7
= x = 1

Conclusion

In this article, we have solved the linear equation $2(2x + 1) = 3(3 - x)$ using step-by-step instructions. We started by distributing the numbers outside the parentheses, then simplified the equation by combining like terms. We added $3x$ to both sides of the equation to isolate the variable $x$ , subtracted 2 from both sides to get rid of the constant term, and finally divided both sides by 7 to solve for $x$ . The final solution is $x = 1$ .

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
When distributing numbers outside the parentheses, make sure to multiply each term inside the parentheses by the number outside.
When simplifying the equation, combine like terms to make it easier to work with.
When adding or subtracting terms, make sure to add or subtract the coefficients of the terms, not the terms themselves.

Real-World Applications

Linear equations have numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, linear equations can be used to model the motion of objects, while in engineering, they can be used to design and optimize systems. In economics, linear equations can be used to model the behavior of markets and make predictions about future trends.

Common Mistakes

When solving linear equations, there are several common mistakes that students make. Some of these mistakes include:

Not following the order of operations (PEMDAS)
Not distributing numbers outside the parentheses correctly
Not combining like terms when simplifying the equation
Not adding or subtracting terms correctly

Conclusion

In conclusion, solving linear equations is a crucial skill that is used in various fields such as physics, engineering, and economics. By following the step-by-step instructions outlined in this article, you can solve linear equations with ease. Remember to follow the order of operations (PEMDAS), distribute numbers outside the parentheses correctly, combine like terms when simplifying the equation, and add or subtract terms correctly. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Introduction

In our previous article, we discussed how to solve linear equations using step-by-step instructions. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, and that's where this Q&A guide comes in. In this article, we'll answer 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.

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

A: To determine if an equation is linear or not, you need to look at 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: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

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: How do I distribute numbers outside the parentheses?

A: To distribute numbers outside the parentheses, you need to multiply each term inside the parentheses by the number outside. For example, if you have the expression 2(3x + 4), you would multiply 2 by each term inside the parentheses to get 6x + 8.

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, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to follow the same steps as you would with a linear equation without fractions. However, you may need to multiply both sides of the equation by a common denominator to eliminate the fractions.

Q: What is the final answer to the equation 2(2x + 1) = 3(3 - x)?

A: The final answer to the equation 2(2x + 1) = 3(3 - x) is x = 1.

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 work by plugging the solution back into the original equation to make sure it's true.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, you need to plug the solution back into the original equation and make sure it's true. If the solution satisfies the equation, then it's correct.

Conclusion

In this Q&A guide, we've answered some of the most frequently asked questions about solving linear equations. We hope that this guide has been helpful in clarifying any confusion you may have had about solving linear equations. Remember to always follow the order of operations (PEMDAS), distribute numbers outside the parentheses correctly, and check your work to make sure your solution is correct. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.