Solving For 'a' In The Equation A/3 + A/4 = 14

Hey guys! Let's dive into solving this equation together. This is a classic algebra problem where we need to isolate the variable a. Understanding how to solve these types of equations is super important for more advanced math, so let's break it down step by step. We'll go through each stage, making sure everyone gets it. Remember, math can be fun when you approach it systematically!

Understanding the Equation

The equation we're tackling is:

a/3 + a/4 = 14

This equation involves fractions, which might seem a bit intimidating at first, but don't worry! The main goal here is to find the value of a that makes this equation true. In simpler terms, we need to figure out what number a is so that when we divide it by 3, and then divide it by 4, and add the results together, we get 14. Think of it like slicing up a pie – we're trying to figure out the original size of the pie based on the sizes of the slices we have.

The key here is to remember that when we're dealing with equations, whatever we do to one side, we must also do to the other side to keep things balanced. This principle is fundamental to solving any algebraic equation. We'll use this principle throughout our solution. Our primary tool in this algebraic adventure is to maintain balance and clarity, ensuring that each step logically leads us closer to the value of a. Mastering this balance is crucial not just for this problem but for tackling a wide range of mathematical challenges. So, let's keep this balance in mind as we proceed, and you'll see how smoothly we can navigate through this equation.

Step 1: Finding a Common Denominator

Fractions can be a bit tricky to work with when they have different denominators (the bottom number). To add a/3 and a/4, we need to find a common denominator. The easiest way to do this is to find the least common multiple (LCM) of 3 and 4. The multiples of 3 are 3, 6, 9, 12, 15..., and the multiples of 4 are 4, 8, 12, 16.... The smallest number that appears in both lists is 12. So, 12 is our least common multiple.

Now, we need to convert both fractions to have a denominator of 12. To do this, we multiply the numerator (top number) and the denominator of each fraction by the number that will make the denominator 12.

• For a/3, we multiply both the numerator and denominator by 4: (a * 4) / (3 * 4) = 4a/12
• For a/4, we multiply both the numerator and denominator by 3: (a * 3) / (4 * 3) = 3a/12

So, our equation now looks like this:

4a/12 + 3a/12 = 14

Finding a common denominator is a foundational step when adding or subtracting fractions. This step allows us to combine the fractions into a single term, making the equation simpler to solve. Think of it like trying to add apples and oranges – you can't directly add them until you have a common unit, like "pieces of fruit." In our case, the common unit is a fraction with a denominator of 12. By converting both fractions to have the same denominator, we're setting the stage for the next step: combining the terms with a.

Step 2: Combining Like Terms

Now that our fractions have a common denominator, we can easily add them together. We have 4a/12 and 3a/12. Since they both have the same denominator, we simply add the numerators (the top numbers) and keep the denominator the same.

(4a + 3a) / 12 = 14

Combining 4a and 3a gives us 7a. So, our equation simplifies to:

7a/12 = 14

Combining like terms is a crucial step in simplifying algebraic equations. This process allows us to consolidate similar elements, making the equation easier to handle. In our case, we had two fractions with the same denominator, which made it straightforward to add their numerators. Think of it like gathering all the same types of objects in a room – you group all the chairs together, all the tables together, and so on. This makes it easier to see what you have and how to work with it. By combining 4a and 3a into 7a, we've reduced the complexity of the equation, bringing us closer to isolating a and finding its value. This step highlights the power of simplification in problem-solving – by making things less complicated, we make them easier to solve.

Step 3: Isolating 'a'

Our goal is to get a by itself on one side of the equation. We currently have 7a/12 = 14. To isolate a, we need to get rid of the 12 in the denominator and the 7 multiplying a.

First, let's get rid of the 12. Since a is being divided by 12, we can do the opposite operation – multiply both sides of the equation by 12. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.

(7a/12) * 12 = 14 * 12

The 12 in the denominator on the left side cancels out with the 12 we're multiplying by, leaving us with:

7a = 168

Now, we have 7a = 168. To isolate a, we need to get rid of the 7 that's multiplying it. We do this by dividing both sides of the equation by 7:

7a / 7 = 168 / 7

The 7 on the left side cancels out, leaving us with a by itself:

a = 24

Isolating the variable is the heart of solving algebraic equations. This process involves performing operations to get the variable (in our case, a) alone on one side of the equation. It's like peeling away the layers of an onion, one step at a time, until you get to the core. We first tackled the division by 12 by multiplying both sides by 12, effectively undoing the division. Then, we addressed the multiplication by 7 by dividing both sides by 7, undoing the multiplication. Each step is the inverse of the previous operation, carefully chosen to eliminate the numbers surrounding a. This methodical approach is essential for solving not just this equation, but any equation where you need to find the value of a variable.

Step 4: Checking the Solution

It's always a good idea to check our answer to make sure it's correct. To do this, we substitute our value for a (which is 24) back into the original equation:

a/3 + a/4 = 14

Substitute a = 24:

24/3 + 24/4 = 14

Simplify:

8 + 6 = 14

14 = 14

The equation holds true! This means our solution, a = 24, is correct.

Checking your solution is a critical step in the problem-solving process. It's like proofreading a document before you submit it – you want to make sure everything is correct. By substituting our calculated value of a back into the original equation, we're essentially verifying that our solution makes the equation true. This step not only confirms that we've arrived at the correct answer but also helps us catch any potential errors we might have made along the way. The satisfaction of seeing both sides of the equation balance out is a rewarding moment, knowing you've successfully solved the problem.

Conclusion

So, we've successfully solved the equation a/3 + a/4 = 14 and found that a = 24. We did this by:

1. Finding a common denominator.
2. Combining like terms.
3. Isolating a.
4. Checking our solution.

Remember, the key to solving equations is to stay organized, show your work, and double-check your answers. You guys got this! Keep practicing, and you'll become equation-solving pros in no time. Math might seem daunting at first, but with a systematic approach and a bit of practice, you can conquer any equation that comes your way. And remember, each problem you solve is a step forward in your mathematical journey. So, keep stepping, keep solving, and most importantly, keep enjoying the process!