Solve The Equation:${ -(6x + 4) - 2x = 2(x + 5) - 1 }$

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, step by step, to help you understand the process and build your confidence in tackling similar problems.

The Equation

The equation we will be solving is:

${ -(6x + 4) - 2x = 2(x + 5) - 1 }$

This equation appears complex, but don't worry, we will break it down into manageable steps.

Step 1: Simplify the Left Side of the Equation

To simplify the left side of the equation, we need to apply the distributive property and combine like terms.

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

Using the distributive property, we can rewrite the equation as:

${ -6x - 4 - 2x = -8x - 4 }$

So, the simplified left side of the equation is:

${ -8x - 4 }$

Step 2: Simplify the Right Side of the Equation

Now, let's simplify the right side of the equation.

${ 2(x + 5) - 1 = 2x + 10 - 1 }$

Using the distributive property, we can rewrite the equation as:

${ 2x + 10 - 1 = 2x + 9 }$

So, the simplified right side of the equation is:

${ 2x + 9 }$

Step 3: Set the Two Sides Equal to Each Other

Now that we have simplified both sides of the equation, we can set them equal to each other.

${ -8x - 4 = 2x + 9 }$

Step 4: Add 8x to Both Sides of the Equation

To isolate the variable x, we need to add 8x to both sides of the equation.

${ -8x + 8x - 4 = 2x + 8x + 9 }$

This simplifies to:

${ -4 = 10x + 9 }$

Step 5: Subtract 9 from Both Sides of the Equation

Next, we need to subtract 9 from both sides of the equation.

${ -4 - 9 = 10x + 9 - 9 }$

This simplifies to:

${ -13 = 10x }$

Step 6: Divide Both Sides of the Equation by 10

Finally, we need to divide both sides of the equation by 10 to solve for x.

${ \frac{-13}{10} = \frac{10x}{10} }$

This simplifies to:

${ x = -\frac{13}{10} }$

Conclusion

And there you have it! We have solved the linear equation step by step, and the solution is x = -\frac{13}{10}. This process may seem daunting at first, but with practice and patience, you will become proficient in solving linear equations.

Tips and Tricks

• Always start by simplifying both sides of the equation.
• Use the distributive property to expand expressions.
• Combine like terms to simplify the equation.
• Add or subtract the same value to both sides of the equation to isolate the variable.
• Divide both sides of the equation by a non-zero value to solve for the variable.

Practice Problems

Try solving the following linear equations on your own:

1. 2x + 5 = 3x - 2
2. x - 3 = 2x + 1
3. 4x + 2 = 2x - 1

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.

Conclusion

Introduction

In our previous article, we covered the step-by-step process of solving linear equations. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, doubts, or need further clarification on certain concepts. That's why we've put together this Q&A guide to help you better understand linear equations and how to solve them.

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's an equation that can be written in the form ax + b = c, where a, b, and c are constants.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 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 distributive property?

A: The distributive property is a rule that allows you to multiply a single term to multiple terms inside parentheses. For example, 2(x + 3) = 2x + 6.

Q: How do I simplify an equation?

A: To simplify an equation, combine like terms and use the distributive property to expand expressions.

Q: What is a like term?

A: A like term is a term that has the same variable and exponent. For example, 2x and 4x are like terms, while 2x and 3y are not.

Q: How do I isolate the variable?

A: To isolate the variable, add or subtract the same value to both sides of the equation, and then divide both sides by a non-zero value.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when working with expressions. The order of operations is:

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 solve a linear equation with fractions?

A: To solve a linear equation with fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 2 and 3 is 6.

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

A: To solve a linear equation with decimals, multiply both sides of the equation by 10 to eliminate the decimals.

Conclusion

We hope this Q&A guide has helped you better understand linear equations and how to solve them. Remember to practice, practice, practice, and don't be afraid to ask for help if you need it. With time and effort, you'll become proficient in solving linear equations and be able to apply them to real-world problems.

Practice Problems

Try solving the following linear equations on your own:

1. 2x + 5 = 3x - 2
2. x - 3 = 2x + 1
3. 4x + 2 = 2x - 1

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.

Conclusion

In conclusion, solving linear equations is a fundamental skill that requires practice and patience. By following the steps outlined in this article and Q&A guide, you can solve linear equations with confidence. Remember to simplify both sides of the equation, use the distributive property, combine like terms, and isolate the variable. With practice, you will become proficient in solving linear equations and be able to apply them to real-world problems.