Solve For X:${3(2-x) + 5 = 2x + 6 - 5x}$

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 linear equation, which is given as:

${3(2-x) + 5 = 2x + 6 - 5x}$

We will break down the solution step by step, using a combination of algebraic manipulations and logical reasoning. By the end of this article, you will have a clear understanding of how to solve linear equations and be able to apply this skill to a wide range of problems.

Understanding the Equation

Before we dive into the solution, let's take a closer look at the equation and understand what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, x) is 1. The equation is also a quadratic equation, which means it can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

In this case, the equation can be rewritten as:

${3(2-x) + 5 = 2x + 6 - 5x}$

This equation represents a relationship between two variables, x and y. The left-hand side of the equation represents the value of y, while the right-hand side represents the value of x.

Step 1: Distribute the Coefficients

The first step in solving the equation is to distribute the coefficients on the left-hand side. This involves multiplying each term inside the parentheses by the coefficient outside the parentheses.

${3(2-x) + 5 = 2x + 6 - 5x}$

Distributing the coefficients, we get:

${6 - 3x + 5 = 2x + 6 - 5x}$

Step 2: Combine Like Terms

The next step is to combine like terms on both sides of the equation. Like terms are terms that have the same variable and exponent. In this case, we can combine the constant terms on the left-hand side and the x terms on the right-hand side.

${6 - 3x + 5 = 2x + 6 - 5x}$

Combining like terms, we get:

${11 - 3x = -3x + 6}$

Step 3: Add 3x to Both Sides

The next step is to add 3x to both sides of the equation. This will help us isolate the x term on one side of the equation.

${11 - 3x = -3x + 6}$

Adding 3x to both sides, we get:

${11 = 6}$

Step 4: Subtract 6 from Both Sides

The next step is to subtract 6 from both sides of the equation. This will help us isolate the constant term on one side of the equation.

${11 = 6}$

Subtracting 6 from both sides, we get:

${5 = 0}$

Step 5: Solve for x

The final step is to solve for x. In this case, we have a contradiction, which means that the equation has no solution.

${5 = 0}$

This means that the equation is inconsistent, and there is no value of x that satisfies the equation.

Conclusion

In this article, we solved a linear equation using a combination of algebraic manipulations and logical reasoning. We broke down the solution into five steps, each of which involved a specific algebraic operation. By the end of this article, you should have a clear understanding of how to solve linear equations and be able to apply this skill to a wide range of problems.

Common Mistakes to Avoid

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

• Not distributing coefficients: Failing to distribute coefficients can lead to incorrect solutions.
• Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
• Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can lead to incorrect solutions.
• Not checking for contradictions: Failing to check for contradictions can lead to incorrect solutions.

Real-World Applications

Linear equations have a wide range of real-world applications. These include:

• Physics: Linear equations are used to describe the motion of objects in physics.
• Engineering: Linear equations are used to design and optimize systems in engineering.
• Economics: Linear equations are used to model economic systems and make predictions about economic trends.
• Computer Science: Linear equations are used in computer science to solve problems in fields such as machine learning and data analysis.

Final Thoughts

Introduction

In our previous article, we discussed how to solve linear equations using a combination of algebraic manipulations and logical reasoning. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It can be written in the form of ax + b = c, where a, b, and c are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

1. Distribute the coefficients on the left-hand side.
2. Combine like terms on both sides of the equation.
3. Add or subtract the same value to both sides of the equation.
4. Check for contradictions.

Q: What is a contradiction?

A: A contradiction is a situation where the equation has no solution. This can occur when the equation is inconsistent, meaning that it is impossible to satisfy the equation.

Q: How do I know if an equation is inconsistent?

A: An equation is inconsistent if it leads to a contradiction. For example, if you have the equation 2x = 3x - 2, you can add 2 to both sides to get 2x + 2 = 3x. Then, you can subtract 2x from both sides to get 2 = x. However, this is a contradiction, since 2 is not equal to x. Therefore, the equation 2x = 3x - 2 is inconsistent.

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 coefficients.
• Not combining like terms.
• Not adding or subtracting the same value to both sides.
• Not checking for contradictions.

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

A: Linear equations have a wide range of real-world applications. Some examples include:

• Physics: Linear equations are used to describe the motion of objects in physics.
• Engineering: Linear equations are used to design and optimize systems in engineering.
• Economics: Linear equations are used to model economic systems and make predictions about economic trends.
• Computer Science: Linear equations are used in computer science to solve problems in fields such as machine learning and data analysis.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

• Read the equation carefully and understand what it represents.
• Use algebraic manipulations to simplify the equation.
• Check for contradictions and inconsistencies.
• Use logical reasoning to solve the equation.

Q: How do I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

• Working through practice problems in a textbook or online resource.
• Using online tools and calculators to solve equations.
• Creating your own practice problems and solving them.
• Joining a study group or working with a tutor to practice solving linear equations.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving linear equations. We covered topics such as what a linear equation is, how to solve a linear equation, and common mistakes to avoid. We also discussed real-world applications of linear equations and provided tips for practicing solving linear equations. By following these tips and practicing regularly, you will become proficient in solving linear equations and be able to apply this skill to a wide range of problems.