Solving Fraction Equations A Step-by-Step Guide For A

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Hey guys! Today, we're diving into the exciting world of algebra, where we'll tackle an equation involving fractions and solve for the elusive variable 'a'. Don't worry if fractions seem intimidating at first; we'll break it down step by step, making it super easy to understand. So, buckle up, grab your thinking caps, and let's get started!

The Equation at Hand

Our mission, should we choose to accept it (and we do!), is to solve for 'a' in the following equation:

4/7 a - 2/3 = -1/3

This might look like a jumble of numbers and letters, but fear not! We'll conquer it with our problem-solving skills. We need to isolate 'a' on one side of the equation. Think of it like a puzzle – we need to carefully move the pieces around until we reveal the value of 'a'.

Step 1: Eliminating the Fractions - Finding the Common Denominator

Fractions can sometimes be a bit tricky to work with directly, so our first move is to eliminate them. To do this, we'll employ a clever technique: multiplying both sides of the equation by the least common denominator (LCD) of all the fractions involved. But what's the LCD, you ask? It's simply the smallest number that all the denominators (the bottom numbers of the fractions) divide into evenly.

In our equation, the denominators are 7 and 3. The LCD of 7 and 3 is 21 (since 21 is the smallest number that both 7 and 3 divide into). Now, we'll multiply every term in the equation by 21. This is crucial – we need to multiply each term to maintain the balance of the equation. It's like a scale; if you add something to one side, you need to add the same thing to the other side to keep it balanced.

So, we have:

21 * (4/7 a - 2/3) = 21 * (-1/3)

Now, we distribute the 21 to each term inside the parentheses:

21 * (4/7 a) - 21 * (2/3) = 21 * (-1/3)

This simplifies to:

(21/7) * 4a - (21/3) * 2 = (21/3) * -1

And further simplifies to:

3 * 4a - 7 * 2 = 7 * -1

Which gives us:

12a - 14 = -7

Look at that! The fractions are gone, and we have a much simpler equation to work with. This is the power of the LCD – it clears the fractions and makes our lives easier.

Step 2: Isolating the Variable Term - Moving the Constants

Our next goal is to isolate the term with 'a' (which is 12a) on one side of the equation. To do this, we need to get rid of the constant term (-14) on the left side. We can do this by adding 14 to both sides of the equation. Remember, we always need to perform the same operation on both sides to maintain balance.

So, we add 14 to both sides:

12a - 14 + 14 = -7 + 14

This simplifies to:

12a = 7

We're getting closer! The 'a' term is almost by itself.

Step 3: Solving for 'a' - Dividing to Isolate

Now, we have 12a = 7. To finally solve for 'a', we need to get 'a' completely alone. Since 'a' is being multiplied by 12, we'll do the opposite operation: divide both sides of the equation by 12.

12a / 12 = 7 / 12

This simplifies to:

a = 7/12

Eureka! We've found the value of 'a'.

The Solution: a = 7/12

Therefore, the solution to the equation 4/7 a - 2/3 = -1/3 is a = 7/12. This is a proper fraction (the numerator is smaller than the denominator), and it's already in its simplest form (7 and 12 have no common factors other than 1).

Step 4: Verification (Optional but Recommended)

To be absolutely sure we've got the right answer, we can plug our solution (a = 7/12) back into the original equation and see if it holds true. This is a great way to catch any mistakes and build confidence in your solution.

Let's substitute a = 7/12 into the original equation:

4/7 * (7/12) - 2/3 = -1/3

First, we multiply the fractions:

(4 * 7) / (7 * 12) - 2/3 = -1/3

28/84 - 2/3 = -1/3

Now, we simplify the first fraction by dividing both the numerator and denominator by their greatest common factor, which is 28:

1/3 - 2/3 = -1/3

Now, we subtract the fractions on the left side (since they have a common denominator):

(1 - 2) / 3 = -1/3

-1/3 = -1/3

Success! The equation holds true. This confirms that our solution, a = 7/12, is correct. Pat yourselves on the back, guys – you've nailed it!

Key Takeaways: Mastering Fraction Equations

Solving equations with fractions might seem daunting at first, but with a systematic approach, it becomes much more manageable. Let's recap the key steps we took:

  1. Eliminate Fractions with the LCD: Find the least common denominator of all the fractions in the equation and multiply both sides of the equation by it. This clears the fractions and simplifies the equation.
  2. Isolate the Variable Term: Use addition or subtraction to move the constant terms away from the term containing the variable.
  3. Solve for the Variable: Use multiplication or division to isolate the variable and find its value.
  4. Verify Your Solution (Optional but Recommended): Substitute your solution back into the original equation to check if it holds true.

Pro Tips for Fraction Equation Ninjas

  • Stay Organized: Keep your work neat and organized to avoid errors. Write each step clearly and align the equal signs.
  • Double-Check Your Work: It's always a good idea to double-check your calculations, especially when working with fractions.
  • Simplify Fractions: Before and after performing operations, simplify fractions whenever possible to make the numbers smaller and easier to work with.
  • Practice Makes Perfect: The more you practice solving fraction equations, the more confident and proficient you'll become.

Real-World Applications: Where Fraction Equations Shine

You might be wondering,