Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of algebra and learn how to solve equations like pros. We'll specifically tackle the equation $\frac{x}{4} + \frac{4x}{12} = \frac{x}{36}$. Don't worry if it looks a bit intimidating at first; we'll break it down step by step and make sure you understand every bit of it. We will also learn how to check the solution(s) to make sure they are correct. So grab your pencils and let's get started. This guide will walk you through the entire process, from simplifying the equation to verifying our solution.

Understanding the Basics of Equation Solving

Alright, before we jump into our specific equation, let's refresh some fundamental concepts. An equation is simply a mathematical statement that asserts the equality of two expressions. It's like a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. The goal when solving an equation is to isolate the variable (in our case, 'x') on one side of the equation. This means getting 'x' all by itself, so we can find its value. To do this, we use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. If you see an 'x' being added to something, you subtract to isolate it. If it's being multiplied, you divide. Keep in mind that when we're dealing with fractions, things can look a little different, but the core principles remain the same. The first thing that you need to do is to find the least common denominator to simplify the equation. With the concept of inverse operations, you can easily get the answer. This equation is not that hard, you will be able to solve it in a few minutes. I will guide you through the process, but you will be the one who will master the algebra knowledge.

Key Concepts and Terminology

• Variable: A symbol (usually a letter, like 'x') that represents an unknown value. This is what we're trying to solve for!
• Coefficient: The number multiplied by the variable (e.g., in 4x, the coefficient is 4).
• Constant: A number that stands alone (e.g., 5 in the equation x + 5 = 10).
• Inverse Operations: Operations that undo each other (addition/subtraction, multiplication/division). This is super important!
• Isolate the Variable: Get the variable by itself on one side of the equation. It's the key to solving the equation!
• Least Common Denominator (LCD): The smallest number that is a multiple of all the denominators in your equation. This is going to be super useful in the current problem. Find the LCD and multiply by it to simplify the equation.

Step-by-Step Solution of the Equation

Now, let's solve the equation $\frac{x}{4} + \frac{4x}{12} = \frac{x}{36}$.

Step 1: Simplify the Fractions

First, let's see if we can simplify any of the fractions. We have $\frac{4x}{12}$. Both 4 and 12 are divisible by 4. So we can simplify this fraction to $\frac{x}{3}$. Now the equation looks like this: $\frac{x}{4} + \frac{x}{3} = \frac{x}{36}$. Now this is easier to handle, and let's continue. We can do it!

Step 2: Find the Least Common Denominator (LCD)

Next, we need to find the least common denominator (LCD) of the fractions. The denominators are 4, 3, and 36. To find the LCD, we look for the smallest number that all three denominators divide into evenly. In this case, the LCD is 36. Because both 4 and 3 divides into 36.

Step 3: Multiply by the LCD

Now we're going to multiply every term in the equation by the LCD (36). This will get rid of the fractions, making the equation much easier to solve. So: 36 * (\frac{x}{4}) + 36 * (\frac{x}{3}) = 36 * (\frac{x}{36}). Let's simplify:

• 36 * (\frac{x}{4}) = 9x
• 36 * (\frac{x}{3}) = 12x
• 36 * (\frac{x}{36}) = x

Now our equation looks like this: 9x + 12x = x. See? No more fractions! This is a good time to cheer up.

Step 4: Combine Like Terms

On the left side of the equation, we have two terms with 'x': 9x and 12x. We can combine these like terms by adding their coefficients: 9x + 12x = 21x. So now our equation is: 21x = x. The most important step of all the process.

Step 5: Isolate the Variable

Our goal is to get 'x' by itself. To do this, we need to move the 'x' term on the right side to the left side. So, we subtract 'x' from both sides of the equation: 21x - x = x - x. This simplifies to: 20x = 0.

Step 6: Solve for x

Finally, we want to solve for 'x'. To do this, we need to get rid of the coefficient of 'x', which is 20. We do this by dividing both sides of the equation by 20: (20x) / 20 = 0 / 20. This simplifies to x = 0. Therefore, the solution to the equation is x = 0.

Checking the Solution

Woohoo! We've found a solution. But is it correct? It's always a good idea to check the solution to make sure we didn't make any mistakes along the way. We do this by plugging the value of 'x' we found back into the original equation and seeing if it makes the equation true.

Substituting the Solution

Our original equation was $\frac{x}{4} + \frac{4x}{12} = \frac{x}{36}$. We found that x = 0. So, let's substitute 0 for 'x' in the equation: $\frac{0}{4} + \frac{4 * 0}{12} = \frac{0}{36}$.

Simplifying and Verifying

Now, let's simplify each part of the equation:

• $\frac{0}{4} = 0$
• $\frac{4 * 0}{12} = \frac{0}{12} = 0$
• $\frac{0}{36} = 0$

So, our equation becomes: 0 + 0 = 0. And, of course, 0 = 0. This is a true statement! This means that our solution, x = 0, is correct. Nice work, guys!

Tips for Success in Solving Equations

• Practice, practice, practice! The more you solve equations, the better you'll become.
• Show your work. Write down every step, especially when you're starting out. This helps you catch mistakes and understand the process better.
• Check your answers. Always substitute your solution back into the original equation to verify that it's correct.
• Don't be afraid to ask for help. If you get stuck, ask your teacher, a friend, or use online resources for help.
• Take your time. Don't rush through the steps. Solving equations takes patience and precision.
• Master the basics. Make sure you understand fundamental concepts like inverse operations, combining like terms, and fractions before tackling more complex equations.

Common Mistakes to Avoid

• Forgetting to do the same thing to both sides of the equation. Remember the balance! Whatever you do to one side, you must do to the other.
• Making arithmetic errors. Double-check your calculations, especially when dealing with fractions or negative numbers.
• Not simplifying fractions. Always simplify fractions to make the equation easier to work with.
• Forgetting the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Not checking your solution. This is a critical step to ensure your answer is correct.

Conclusion: You've Got This!

Congratulations! You've successfully solved an equation and checked your solution. Remember, solving equations is a fundamental skill in algebra, and with practice, you'll become a pro. Keep practicing, stay patient, and don't hesitate to ask for help. You've got this, and with enough practice, you'll be solving equations like a boss in no time! Keep going! And always, always check your solutions!