Solve The Equation: $\frac{1}{2}(-10x + 4) = -4(-3 + 2x) + 8$Enter Your Work In The Space Provided.Enter The Answer In The Space Provided. Use Numbers Instead Of Words.

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, $\frac{1}{2}(-10x + 4) = -4(-3 + 2x) + 8$. We will break down the solution into manageable steps, making it easy for readers to understand and follow along.

Step 1: Distribute and Simplify

The first step in solving the equation is to distribute and simplify both sides of the equation. This involves multiplying the numbers outside the parentheses by the terms inside.

$\frac{1}{2}(-10x + 4) = -4(-3 + 2x) + 8$

Distributing the numbers outside the parentheses, we get:

$-5x + 2 = 12 - 8x + 8$

Simplifying the equation further, we get:

$-5x + 2 = 20 - 8x$

Step 2: Isolate the Variable

The next step is to isolate the variable x by getting all the terms with x on one side of the equation and the constant terms on the other side.

$-5x + 2 = 20 - 8x$

Adding 8x to both sides of the equation, we get:

$3x + 2 = 20$

Subtracting 2 from both sides of the equation, we get:

$3x = 18$

Step 3: Solve for x

The final step is to solve for x by dividing both sides of the equation by the coefficient of x.

$3x = 18$

Dividing both sides of the equation by 3, we get:

$x = 6$

Conclusion

In this article, we solved the linear equation $\frac{1}{2}(-10x + 4) = -4(-3 + 2x) + 8$ using a step-by-step approach. We distributed and simplified the equation, isolated the variable x, and finally solved for x. The solution to the equation is x = 6.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that the equation is simplified correctly.
• Isolating the variable x is a crucial step in solving linear equations. Make sure to get all the terms with x on one side of the equation and the constant terms on the other side.
• When solving for x, make sure to divide both sides of the equation by the coefficient of x to avoid any errors.

Common Mistakes to Avoid

• Not following the order of operations (PEMDAS) can lead to incorrect solutions.
• Failing to isolate the variable x can make it difficult to solve the equation.
• Not checking the solution by plugging it back into the original equation can lead to incorrect answers.

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 solved the linear equation $\frac{1}{2}(-10x + 4) = -4(-3 + 2x) + 8$ using a step-by-step approach. In this article, we will answer some frequently asked questions about solving linear equations.

Q: What is a linear equation?

A linear equation is an equation in which the highest power of the variable (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: How do I solve a linear equation?

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

1. Distribute and simplify the equation.
2. Isolate the variable x by getting all the terms with x on one side of the equation and the constant terms on the other side.
3. Solve for x by dividing both sides of the equation by the coefficient of x.

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

The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:

• P: Parentheses (evaluate expressions inside parentheses first)
• E: Exponents (evaluate any exponential expressions next)
• M: Multiplication and Division (evaluate any multiplication and division operations from left to right)
• A: Addition and Subtraction (finally, evaluate any addition and subtraction operations from left to right)

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

To determine if an equation is linear or not, you need to look at the highest power of the variable (x). If the highest power of x is 1, then the equation is linear. If the highest power of x is greater than 1, then the equation is not linear.

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

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 correct.

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

Some common mistakes to avoid when solving linear equations include:

• Not following the order of operations (PEMDAS)
• Failing to isolate the variable x
• Not checking the solution by plugging it back into the original equation

Q: How do I check my work when solving linear equations?

To check your work when solving linear equations, you need to plug the solution back into the original equation and make sure it's true. This will help you catch any mistakes you may have made.

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

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

Solving linear equations is a fundamental skill that has numerous real-world applications. By following a step-by-step approach and avoiding common mistakes, students can master the art of solving linear equations. Whether it's in physics, engineering, or economics, linear equations play a crucial role in understanding and predicting complex systems.