Solve The Equation:${ 5 - 2(3 - X) = 4x + 10 }$

Mar 13, 2025 by ADMIN 49 views

Solving the Equation: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of algebra and focus on solving a linear equation. The equation we will be working with is $5 - 2(3 - x) = 4x + 10$ . This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what we're dealing with. The equation is a linear equation, which means it can be written in the form $ax + b = c$ , where $a$ , $b$ , and $c$ are constants. In this case, our equation is $5 - 2(3 - x) = 4x + 10$ . Our goal is to isolate the variable $x$ and find its value.

Distributing the Negative Sign

To start solving the equation, we need to distribute the negative sign inside the parentheses. This means we need to multiply the negative sign by each term inside the parentheses. So, $-2(3 - x)$ becomes $-6 + 2x$ . Now our equation looks like this: $5 - 6 + 2x = 4x + 10$ .

Combining Like Terms

Next, we need to combine like terms on both sides of the equation. On the left side, we have $5 - 6$ , which simplifies to $-1$ . On the right side, we have $4x + 10$ . Now our equation looks like this: $-1 + 2x = 4x + 10$ .

Isolating the Variable

Now that we have combined like terms, we can start isolating the variable $x$ . To do this, we need to get all the terms with $x$ on one side of the equation. We can do this by subtracting $2x$ from both sides of the equation. This gives us $-1 = 2x + 10$ .

Simplifying the Equation

Next, we need to simplify the equation by getting rid of the constant term on the right side. We can do this by subtracting $10$ from both sides of the equation. This gives us $-11 = 2x$ .

Solving for x

Now that we have simplified the equation, we can solve for $x$ . To do this, we need to divide both sides of the equation by $2$ . This gives us $x = -\frac{11}{2}$ .

Conclusion

In this article, we have solved the equation $5 - 2(3 - x) = 4x + 10$ . We started by distributing the negative sign inside the parentheses, then combined like terms, isolated the variable, simplified the equation, and finally solved for $x$ . With these steps, we have found the solution to the equation.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS).
Make sure to distribute the negative sign correctly when working with parentheses.
Combining like terms can help simplify the equation and make it easier to solve.
Isolating the variable is a crucial step in solving linear equations.
Simplifying the equation can help you find the solution more easily.

Real-World Applications

Solving linear equations has many real-world applications. For example, in physics, linear equations are used to describe the motion of objects. In economics, linear equations are used to model the behavior of markets. In computer science, linear equations are used to solve systems of equations and find the solution to complex problems.

Common Mistakes

When solving linear equations, there are several common mistakes to avoid. These include:

Not distributing the negative sign correctly
Not combining like terms
Not isolating the variable
Not simplifying the equation
Not following the order of operations (PEMDAS)

Final Thoughts

Solving linear equations is a fundamental skill in mathematics. With practice and patience, anyone can master this skill and apply it to real-world problems. Remember to follow the order of operations, distribute the negative sign correctly, combine like terms, isolate the variable, and simplify the equation. With these steps, you'll be solving linear equations like a pro in no time.

Additional Resources

For more information on solving linear equations, check out the following resources:

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

Conclusion

In conclusion, solving the equation $5 - 2(3 - x) = 4x + 10$ requires a clear understanding of the steps involved. By distributing the negative sign, combining like terms, isolating the variable, simplifying the equation, and solving for $x$ , we can find the solution to the equation. With practice and patience, anyone can master this skill and apply it to real-world problems.

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Introduction

In our previous article, we solved the equation $5 - 2(3 - x) = 4x + 10$ using a step-by-step approach. However, we know that sometimes, the best way to learn is through questions and answers. In this article, we will provide a Q&A guide 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. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when working with mathematical expressions. 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 the negative sign?

A: When working with parentheses, you may need to distribute the negative sign. To do this, multiply the negative sign by each term inside the parentheses. For example, if you have -2(3 - x), you would multiply the negative sign by 3 and -x, resulting in -6 + 2x.

Q: What is the difference between combining like terms and simplifying the equation?

A: Combining like terms involves adding or subtracting terms that have the same variable and coefficient. Simplifying the equation involves getting rid of any constant terms on the right side of the equation.

Q: How do I isolate the variable?

A: To isolate the variable, you need to get all the terms with the variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation.

Q: What is the final step in solving a linear equation?

A: The final step in solving a linear equation is to solve for the variable. This involves dividing both sides of the equation by the coefficient of the variable.

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

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

Not distributing the negative sign correctly
Not combining like terms
Not isolating the variable
Not simplifying the equation
Not following the order of operations (PEMDAS)

Q: How can I practice solving linear equations?

A: There are many resources available to help you practice solving linear equations, including:

Online math websites and apps
Math textbooks and workbooks
Practice problems and worksheets
Online communities and forums

Q: What are some real-world applications of solving linear equations?

A: Solving linear equations has many real-world applications, including:

Physics: Solving linear equations is used to describe the motion of objects.
Economics: Solving linear equations is used to model the behavior of markets.
Computer Science: Solving linear equations is used to solve systems of equations and find the solution to complex problems.

Conclusion

In this Q&A guide, we have provided answers to some of the most common questions about solving linear equations. By following the steps outlined in this guide, you can master the skill of solving linear equations and apply it to real-world problems. Remember to practice regularly and seek help when needed. With patience and persistence, you can become proficient in solving linear equations.