Solving For X: A Step-by-Step Guide

Oct 13, 2025 by ADMIN 36 views

Alright, guys, let's dive into solving a classic algebraic equation! We're tackling: $\frac{x}{4}+x=\frac{x}{2}-3$ . Don't worry; it's not as intimidating as it looks. We'll break it down into manageable steps so everyone can follow along. Understanding how to solve for a variable like 'x' is super important in math, science, and even everyday problem-solving. So, buckle up, and let's get started!

Clearing Fractions: The First Hurdle

Our main goal when you see fractions is usually to get rid of them as quickly as possible. Fractions can make equations look messy, but there's a neat trick to eliminate them: multiplying every term in the equation by the least common multiple (LCM) of the denominators. In our case, the denominators are 4 and 2. The LCM of 4 and 2 is simply 4. So, we're going to multiply every single term in the equation by 4. This means we'll multiply $\frac{x}{4}$ by 4, x by 4, $\frac{x}{2}$ by 4, and -3 by 4. Doing this carefully ensures we maintain the equality of the equation.

Let's walk through it. Multiplying $\frac{x}{4}$ by 4 gives us just x. Multiplying x by 4 gives us 4x. On the right side, multiplying $\frac{x}{2}$ by 4 gives us 2x, and multiplying -3 by 4 gives us -12. So, our equation now looks like this: x + 4x = 2x - 12. Notice how the fractions have completely disappeared! This makes the equation much easier to handle. This step is crucial because it transforms a potentially confusing equation into a simple linear equation that we can solve with basic algebraic manipulations. By clearing the fractions, we pave the way for combining like terms and isolating the variable 'x'.

Combining Like Terms: Simplifying the Equation

Now that we've cleared the fractions, let's simplify the equation further by combining like terms. On the left side of the equation, we have 'x' and '4x'. These are like terms because they both contain the variable 'x' raised to the same power (which is 1 in this case). To combine them, we simply add their coefficients. The coefficient of 'x' is 1 (since x is the same as 1x), and the coefficient of '4x' is 4. Adding these together gives us 1 + 4 = 5. So, x + 4x simplifies to 5x.

Our equation now looks like this: 5x = 2x - 12. Notice how much simpler it is compared to the original equation with fractions! Combining like terms is a fundamental technique in algebra that helps us to reduce the complexity of equations and make them easier to solve. By identifying and combining like terms, we can consolidate multiple terms into a single term, which simplifies the equation and brings us closer to isolating the variable we're trying to solve for. In this case, combining 'x' and '4x' into '5x' makes the equation more manageable and sets us up for the next step of isolating 'x'.

Isolating x: Getting x by Itself

Our next key step is to isolate 'x' on one side of the equation. Currently, we have 'x' terms on both sides: 5x on the left and 2x on the right. To isolate 'x', we need to move all the 'x' terms to one side and all the constant terms to the other side. A common strategy is to move the 'x' terms to the side that already has the larger coefficient. In this case, 5x is greater than 2x, so we'll move the 2x term from the right side to the left side.

To do this, we subtract 2x from both sides of the equation. This ensures that we maintain the equality of the equation. Subtracting 2x from 5x gives us 3x. Subtracting 2x from 2x on the right side cancels out the 'x' term, leaving us with just -12. So, our equation now looks like this: 3x = -12. Notice how all the 'x' terms are now on the left side, and all the constant terms are on the right side. We're one step closer to solving for 'x'!

Now, to completely isolate 'x', we need to get rid of the coefficient 3 that's multiplying it. To do this, we divide both sides of the equation by 3. Dividing 3x by 3 gives us just 'x'. Dividing -12 by 3 gives us -4. So, our final equation is: x = -4. Congratulations! We've successfully isolated 'x' and found its value.

The Solution: x = -4

So, after all that work, we've found that x = -4. That's the solution to our original equation: $\frac{x}{4}+x=\frac{x}{2}-3$ . But, it's always a good idea to check our work to make sure we didn't make any mistakes along the way.

Checking Our Work: Making Sure We're Right

To check our solution, we substitute x = -4 back into the original equation and see if both sides of the equation are equal. The original equation was: $\frac{x}{4}+x=\frac{x}{2}-3$ . Substituting x = -4 gives us: $\frac{-4}{4} + (-4) = \frac{-4}{2} - 3$ .

Let's simplify the left side of the equation. $\frac{-4}{4}$ is equal to -1. Adding -4 to -1 gives us -5. So, the left side of the equation simplifies to -5.

Now, let's simplify the right side of the equation. $\frac{-4}{2}$ is equal to -2. Subtracting 3 from -2 gives us -5. So, the right side of the equation also simplifies to -5.

Since both sides of the equation are equal to -5 when we substitute x = -4, our solution is correct! This confirms that x = -4 is indeed the solution to the original equation. Always remember to check your work, especially in exams, to catch any potential errors and ensure you get the correct answer.

Key Takeaways: Mastering the Process

Let's recap the key steps we took to solve this equation:

Clearing Fractions: Multiply every term in the equation by the least common multiple (LCM) of the denominators to eliminate fractions.
Combining Like Terms: Simplify the equation by combining like terms on both sides.
Isolating x: Move all 'x' terms to one side of the equation and all constant terms to the other side, then divide by the coefficient of 'x' to solve for 'x'.
Checking Your Work: Substitute your solution back into the original equation to ensure both sides are equal.

By following these steps carefully, you can confidently solve a wide variety of algebraic equations. Remember to practice regularly to reinforce your skills and build your confidence. Solving equations is a fundamental skill in mathematics, and mastering it will open doors to more advanced topics and real-world applications.

So there you have it! Solving for 'x' doesn't have to be a mystery. Just remember the steps, practice, and you'll be solving equations like a pro in no time! Keep up the great work, guys!