Solve The Following Equation For $x$.$\frac{3}{9}(x+24)=12$x=\square$

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, $\frac{3}{9}(x+24)=12$, to find the value of $x$. We will break down the solution into manageable steps, making it easy for readers to understand and follow along.

Understanding the Equation

The given equation is $\frac{3}{9}(x+24)=12$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The equation involves a fraction, which we will simplify before proceeding with the solution.

Step 1: Simplify the Fraction

The fraction $\frac{3}{9}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

$\frac{3}{9} = \frac{1}{3}$

So, the equation becomes:

$\frac{1}{3}(x+24)=12$

Step 2: Multiply Both Sides by 3

To eliminate the fraction, we can multiply both sides of the equation by 3.

$1(x+24)=36$

Step 3: Distribute the 1

Since any number multiplied by 1 is itself, we can simplify the equation by distributing the 1.

$x+24=36$

Step 4: Subtract 24 from Both Sides

To isolate the variable $x$, we need to get rid of the constant term 24. We can do this by subtracting 24 from both sides of the equation.

$x=12$

Conclusion

In this article, we solved the linear equation $\frac{3}{9}(x+24)=12$ to find the value of $x$. By simplifying the fraction, multiplying both sides by 3, distributing the 1, and subtracting 24 from both sides, we arrived at the solution $x=12$. This problem demonstrates the importance of following the order of operations and using algebraic properties to solve linear equations.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
• Simplifying fractions and using algebraic properties can make solving linear equations much easier.
• Practice solving linear equations with different coefficients and constants to become more comfortable with the process.

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.

Common Mistakes to Avoid

• Not following the order of operations (PEMDAS) can lead to incorrect solutions.
• Failing to simplify fractions and use algebraic properties can make solving linear equations more difficult.
• Not checking the solution by plugging it back into the original equation can lead to incorrect answers.

Conclusion

Introduction

In our previous article, we solved the linear equation $\frac{3}{9}(x+24)=12$ to find the value of $x$. However, we know that practice makes perfect, and solving linear equations can be a challenging task for many students. 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 linear equation is an equation in which the highest power of the variable (usually $x$) is 1. In other words, a linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

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

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

• The highest power of the variable is 1.
• The equation can be written in the form $ax + b = c$.
• The equation does not involve any exponents or roots.

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

The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when solving an equation. The acronym 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 simplify fractions in linear equations?

To simplify fractions in linear equations, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.
• Simplify the resulting fraction.

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

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. In other words, a linear equation is an equation that can be written in the form $ax + b = c$, while a quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$.

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

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

• Simplify the fraction by dividing both the numerator and denominator by their GCD.
• Multiply both sides of the equation by the denominator to eliminate the fraction.
• Distribute the multiplication to the terms inside the parentheses.
• Combine like terms and solve for the variable.

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 simplify fractions and use algebraic properties.
• Not checking the solution by plugging it back into the original equation.

Conclusion

Solving linear equations can be a challenging task, but with practice and patience, anyone can become proficient. By following the steps outlined in this article and avoiding common mistakes, students can become more confident in their ability to solve linear equations. Remember to practice regularly and seek help when needed to become a proficient problem-solver.

Additional Resources

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

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

Practice Problems

Try solving the following linear equations:

1. $\frac{2}{3}(x-5)=10$
2. $x+\frac{1}{4}=12$
3. $\frac{3}{5}x=15$

Answer Key

1. $x=20$
2. $x=\frac{47}{4}$
3. $x=25$