Solving Equations: Common Denominator Method

Hey guys! Today, we're diving into a common and super useful technique in algebra: solving equations by multiplying both sides by the common denominator. This method is especially handy when you're dealing with equations that have fractions, as it helps clear those fractions out and makes the equation much easier to handle. We'll be focusing on an example equation to illustrate this process step-by-step. So, buckle up, and let's get started!

The Equation: A Quick Look

Let's take a look at the equation we're going to solve:

$\frac{5}{2x} = \frac{2}{x} - \frac{1}{12}$

As you can see, we've got fractions with variables in the denominators. This is where the common denominator method shines. Our main goal here is to eliminate these fractions to simplify the equation. To kick things off, we need to figure out what the common denominator actually is. The common denominator is essentially the smallest multiple that all the denominators in the equation share. It's like finding the sweet spot that all fractions can comfortably convert to.

Finding the Common Denominator

Okay, so how do we find this magical common denominator? Well, we need to look at each denominator individually and consider their factors. In our equation, the denominators are 2x, x, and 12. To find the common denominator, we need to identify the least common multiple (LCM) of these expressions. Think of it like this: we need a number that each of these denominators can divide into evenly. The first denominator is 2x, which means our common denominator must be a multiple of both 2 and x. The second denominator is x, which is already accounted for in 2x. The third denominator is 12, which can be factored into 2 * 2 * 3. So, we need to make sure our common denominator includes these factors as well. Considering all these factors, the least common multiple – our common denominator – is 12x. This is because 12x is the smallest expression that is divisible by 2x, x, and 12.

Multiplying Through by 12x

Now that we've identified our common denominator as 12x, the next step is to multiply both sides of the equation by this term. This is a crucial step because it will eliminate the fractions, making the equation much easier to solve. When we multiply each term by 12x, we're essentially scaling up the equation while maintaining the balance between both sides. This technique works because we're performing the same operation on both sides, ensuring that the equality remains true. Let's break down the multiplication process step by step to make it clear.

On the left side of the equation, we have $\frac{5}{2x}$. Multiplying this by 12x gives us:

(12x) * $\frac{5}{2x}$ = $\frac{12x * 5}{2x}$.

Notice that the 'x' in the numerator and denominator cancels out, and 12 divided by 2 is 6. So, this simplifies to:

6 * 5 = 30.

Moving to the right side of the equation, we have two terms: $\frac{2}{x}$ and -$\frac{1}{12}$. Let's multiply each of these by 12x separately. First, we multiply 12x by $\frac{2}{x}$:

(12x) * $\frac{2}{x}$ = $\frac{12x * 2}{x}$.

Again, the 'x' terms cancel out, leaving us with:

12 * 2 = 24.

Next, we multiply 12x by -$\frac{1}{12}$:

(12x) * (-$\frac{1}{12}$) = -$\frac{12x}{12}$.

In this case, the 12s cancel out, and we're left with:

-x.

So, after multiplying each term in the original equation by the common denominator 12x, our equation now looks like this:

30 = 24 - x

Simplifying and Solving for x

With the fractions cleared, our equation is now a straightforward linear equation. To solve for x, we need to isolate x on one side of the equation. We can do this by performing algebraic operations on both sides to maintain the balance. Our current equation is:

30 = 24 - x.

To get x by itself, let's start by subtracting 24 from both sides of the equation. This will move the constant term from the right side to the left side. Subtracting 24 from both sides gives us:

30 - 24 = 24 - x - 24,

which simplifies to:

6 = -x.

Now, we have -x on the right side, but we want to find the value of x, not -x. To do this, we can multiply both sides of the equation by -1. This will change the sign of both terms, giving us the positive value of x. Multiplying both sides by -1, we get:

6 * (-1) = -x * (-1),

which simplifies to:

-6 = x.

So, we've found that x = -6. This is the solution to our equation. To be sure we've got it right, it’s always a good idea to check our solution by plugging it back into the original equation.

Checking the Solution

Alright, we've found our solution, x = -6, but let's not pop the champagne just yet! The best practice in mathematics, especially when dealing with equations, is to verify our solution. This step ensures that our answer is correct and that we haven't made any mistakes along the way. To check our solution, we're going to substitute x = -6 back into the original equation:

$\frac{5}{2x} = \frac{2}{x} - \frac{1}{12}$

Let's plug in x = -6:

$\frac{5}{2(-6)} = \frac{2}{-6} - \frac{1}{12}$

Now, we simplify each side of the equation separately. On the left side, we have:

$\frac{5}{2(-6)} = \frac{5}{-12}$

So, the left side simplifies to -$\frac{5}{12}$.

Now, let's simplify the right side. We have:

$\frac{2}{-6} - \frac{1}{12}$

First, simplify $\frac{2}{-6}$ to -$\frac{1}{3}$. So, the right side becomes:

-\frac{1}{3} - \frac{1}{12}$

To subtract these fractions, we need a common denominator. The common denominator for 3 and 12 is 12. So, we convert -$\frac{1}{3}$ to an equivalent fraction with a denominator of 12. Multiply both the numerator and the denominator by 4:

-\frac{1}{3} = -\frac{1 * 4}{3 * 4} = -\frac{4}{12}$

Now, our right side looks like this:

-\frac{4}{12} - \frac{1}{12}$

Subtracting these fractions gives us:

-\frac{4}{12} - \frac{1}{12} = -\frac{4 + 1}{12} = -\frac{5}{12}$

So, the right side simplifies to -$\frac{5}{12}$.

Now, let's compare both sides of the equation. We found that the left side is -$\frac{5}{12}$ and the right side is also -$\frac{5}{12}$. Since both sides are equal, our solution x = -6 is correct!

The Final Result

We've successfully solved the equation $\frac{5}{2x} = \frac{2}{x} - \frac{1}{12}$ by multiplying both sides by the common denominator. We found that x = -6. This method is a powerful tool for tackling equations with fractions, and as you practice, you'll become even more comfortable with it. Remember, the key is to find that common denominator, multiply it through, and then simplify and solve. And always, always check your solution! You've got this!

Key Takeaways

• Common Denominator: The least common multiple of the denominators in an equation.
• Multiplying Through: Eliminates fractions by multiplying each term by the common denominator.
• Simplifying: Combine like terms and isolate the variable to solve.
• Checking: Always substitute your solution back into the original equation to verify.

Practice Makes Perfect

The best way to master this technique is through practice. Try solving similar equations on your own, and don't hesitate to reach out if you get stuck. Keep up the great work, and happy solving!