Solve For $x$.$\[ \begin{array}{c} 4(8-x) = 24 \\ x = [?] \end{array} \\]

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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 type of linear equation, which is a simple algebraic equation with one variable. We will use the given equation as an example and walk through the step-by-step process of solving for the variable x.

The Given Equation

The given equation is:

4(8−x)=244(8-x) = 24

Our goal is to solve for the variable x.

Step 1: Distribute the Coefficient

To start solving the equation, we need to distribute the coefficient 4 to the terms inside the parentheses. This will give us:

32−4x=2432 - 4x = 24

Step 2: Isolate the Variable Term

Next, we need to isolate the variable term by moving the constant term to the other side of the equation. We can do this by subtracting 32 from both sides of the equation:

−4x=24−32-4x = 24 - 32

Simplifying the right-hand side of the equation, we get:

−4x=−8-4x = -8

Step 3: Solve for the Variable

Now that we have isolated the variable term, we can solve for x by dividing both sides of the equation by -4:

x=−8−4x = \frac{-8}{-4}

Simplifying the right-hand side of the equation, we get:

x=2x = 2

Conclusion

In this article, we have walked through the step-by-step process of solving a linear equation. We started with the given equation, distributed the coefficient, isolated the variable term, and finally solved for the variable x. The solution to the equation is x = 2.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

  • Distribute the coefficient: When you see a coefficient inside parentheses, make sure to distribute it to the terms inside the parentheses.
  • Isolate the variable term: Move the constant term to the other side of the equation to isolate the variable term.
  • Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary parentheses.
  • Check your solution: Once you have solved for the variable, plug the solution back into the original equation to check if it is true.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear equations:

  • Forgetting to distribute the coefficient: Make sure to distribute the coefficient to the terms inside the parentheses.
  • Not isolating the variable term: Move the constant term to the other side of the equation to isolate the variable term.
  • Not simplifying the equation: Simplify the equation by combining like terms and eliminating any unnecessary parentheses.
  • Not checking your solution: Plug the solution back into the original equation to check if it is true.

Real-World Applications

Linear equations have many 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 economic trends.
  • Computer Science: Linear equations are used in computer graphics and game development to create realistic simulations.

Conclusion

Introduction

In our previous article, we walked through the step-by-step process of solving a linear equation. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will address some common questions and concerns that students may have when solving linear equations.

Q: What is a linear equation?

A: A linear equation is a type of algebraic equation that has one variable and is in the form of ax + b = c, where a, b, and c are constants.

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

A: To determine if an equation is linear, look for the following characteristics:

  • The equation has only one variable.
  • The variable is raised to the power of 1 (i.e., it is not squared or cubed).
  • The equation does not contain any fractions or decimals.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation has one variable and is in the form of ax + b = c, while a quadratic equation has two variables and is in the form of ax^2 + bx + c = 0.

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

A: To solve a linear equation with fractions, follow these steps:

  1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
  2. Simplify the equation by combining like terms.
  3. Solve for the variable.

Q: What is the order of operations when solving linear equations?

A: When solving linear equations, follow the order of operations (PEMDAS):

  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 check my solution to a linear equation?

A: To check your solution to a linear equation, plug the solution back into the original equation and simplify. If the equation is true, then your solution is correct.

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

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

  • Forgetting to distribute the coefficient.
  • Not isolating the variable term.
  • Not simplifying the equation.
  • Not checking your solution.

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

A: Linear equations have many 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 economic trends.
  • Computer Science: Linear equations are used in computer graphics and game development to create realistic simulations.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article and addressing common questions and concerns, you can become proficient in solving linear equations and apply them to real-world problems. Remember to practice regularly and seek help when needed. With patience and persistence, you will become a master of linear equations!