Solving The Equation (1/14)(a+8) = (1/4)(3-a) A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of equations and tackling a specific one that might seem a bit tricky at first glance: $\frac{1}{14}(a+8)=\frac{1}{4}(3-a)$. Don't worry, we'll break it down step-by-step, making sure everyone understands the process. Whether you're a student brushing up on your algebra skills or just someone who enjoys a good mathematical puzzle, this guide is for you. We'll not only solve the equation but also discuss the underlying principles and strategies you can use to conquer similar problems. So, grab your pencils, and let's get started!

Understanding the Basics: What is an Equation?

Before we jump into solving our specific equation, let's quickly recap what an equation actually is. Simply put, an equation is a mathematical statement that asserts the equality of two expressions. It's like a balanced scale, where both sides must have the same weight. In our case, the equation $\frac{1}{14}(a+8)=\frac{1}{4}(3-a)$ states that the expression on the left side is equal to the expression on the right side. Our goal is to find the value(s) of the variable 'a' that make this statement true. This value is called the solution to the equation.

Equations can be linear, quadratic, or involve more complex functions. The equation we're dealing with is a linear equation because the variable 'a' appears only to the first power. Solving linear equations typically involves isolating the variable on one side of the equation by performing the same operations on both sides to maintain the balance. We'll use this principle throughout our solution process.

Think of solving an equation like detective work. We have clues (the terms and operations in the equation), and we need to use these clues to find the hidden value of our variable. There are several techniques we can use, such as distributing, combining like terms, and using inverse operations. Let's see how these techniques apply to our equation!

Step-by-Step Solution: Cracking the Code

Okay, let's get down to business and solve the equation $\frac{1}{14}(a+8)=\frac{1}{4}(3-a)$. We'll take it one step at a time to ensure clarity.

Step 1: Clearing the Fractions

The first thing that might jump out at you is the fractions. Fractions can sometimes make equations look intimidating, but don't fret! We can easily get rid of them by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In our case, the denominators are 14 and 4. The LCM of 14 and 4 is 28. So, we'll multiply both sides of the equation by 28.

This LCM trick is a powerful tool because it eliminates the fractions, making the equation much easier to handle. By multiplying both sides by the same number, we maintain the balance of the equation, ensuring that the equality remains true.

So, let's multiply both sides by 28:

$28 * \frac{1}{14}(a+8) = 28 * \frac{1}{4}(3-a)$

This simplifies to:

$2(a+8) = 7(3-a)$

See how much cleaner the equation looks now? We've successfully eliminated the fractions and are ready for the next step.

Step 2: Distributing

Now that we've cleared the fractions, we need to deal with the parentheses. To do this, we'll use the distributive property, which states that a(b + c) = ab + ac. In other words, we multiply the term outside the parentheses by each term inside the parentheses.

On the left side, we have 2(a + 8). Distributing the 2 gives us 2 * a + 2 * 8, which simplifies to 2a + 16.

On the right side, we have 7(3 - a). Distributing the 7 gives us 7 * 3 - 7 * a, which simplifies to 21 - 7a.

So, our equation now looks like this:

$2a + 16 = 21 - 7a$

We've successfully distributed and removed the parentheses. The equation is getting simpler and simpler!

Step 3: Gathering the 'a' Terms

Our next goal is to get all the terms containing 'a' on one side of the equation. It doesn't matter which side we choose, but it's often helpful to choose the side that will result in a positive coefficient for 'a'. In this case, we can add 7a to both sides to move the '-7a' term from the right side to the left side.

Adding 7a to both sides, we get:

$2a + 16 + 7a = 21 - 7a + 7a$

This simplifies to:

$9a + 16 = 21$

We've successfully gathered all the 'a' terms on the left side. Now, we need to isolate 'a' further by getting rid of the constant term on the left side.

Step 4: Isolating the 'a' Term

To isolate the 'a' term, we need to get rid of the +16 on the left side. We can do this by subtracting 16 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance.

Subtracting 16 from both sides, we get:

$9a + 16 - 16 = 21 - 16$

This simplifies to:

$9a = 5$

We're almost there! We've isolated the 'a' term on the left side. Now, we just need to get 'a' by itself.

Step 5: Solving for 'a'

Finally, to solve for 'a', we need to get rid of the 9 that's multiplying it. We can do this by dividing both sides of the equation by 9. This is the inverse operation of multiplication, and it will leave 'a' all by itself.

Dividing both sides by 9, we get:

$\frac{9a}{9} = \frac{5}{9}$

This simplifies to:

$a = \frac{5}{9}$

And there you have it! We've successfully solved the equation. The value of 'a' that makes the equation true is $\frac{5}{9}$.

Checking Our Solution: The Final Step

It's always a good idea to check our solution to make sure we haven't made any mistakes along the way. To do this, we'll substitute our solution, $\frac{5}{9}$, back into the original equation and see if both sides are equal.

Our original equation was:

$\frac{1}{14}(a+8)=\frac{1}{4}(3-a)$

Substituting $a = \frac{5}{9}$, we get:

$\frac{1}{14}(\frac{5}{9}+8)=\frac{1}{4}(3-\frac{5}{9})$

Let's simplify each side separately.

Left Side:

$\frac{1}{14}(\frac{5}{9}+8) = \frac{1}{14}(\frac{5}{9} + \frac{72}{9}) = \frac{1}{14}(\frac{77}{9}) = \frac{77}{126} = \frac{11}{18}$

Right Side:

$\frac{1}{4}(3-\frac{5}{9}) = \frac{1}{4}(\frac{27}{9}-\frac{5}{9}) = \frac{1}{4}(\frac{22}{9}) = \frac{22}{36} = \frac{11}{18}$

Since both sides simplify to $\frac{11}{18}$, our solution is correct! We've verified that $a = \frac{5}{9}$ is indeed the solution to the equation.

Key Takeaways: Mastering Equation Solving

We've successfully navigated through the solution of the equation $\frac{1}{14}(a+8)=\frac{1}{4}(3-a)$. Let's recap the key strategies and concepts we used:

• Clearing Fractions: Multiplying both sides of the equation by the LCM of the denominators is a powerful technique to simplify equations containing fractions. This eliminates the fractions and makes the equation easier to work with.
• Distributing: The distributive property allows us to remove parentheses by multiplying the term outside the parentheses by each term inside. This is a crucial step in simplifying equations.
• Combining Like Terms: Gathering terms with the same variable on one side of the equation and constant terms on the other side helps to isolate the variable we're trying to solve for.
• Inverse Operations: Using inverse operations (addition/subtraction, multiplication/division) allows us to isolate the variable. Remember, whatever operation we perform on one side of the equation, we must perform on the other to maintain the balance.
• Checking the Solution: Always check your solution by substituting it back into the original equation. This helps to catch any errors and ensures that your solution is correct.

These strategies are not just applicable to this specific equation but are fundamental tools in solving a wide range of algebraic equations. Practice using these techniques, and you'll become a pro at equation solving!

Practice Makes Perfect: More Equations to Solve

Now that we've worked through this example together, it's time for you to put your skills to the test! Here are a few similar equations you can try solving on your own:

1. $\frac{1}{6}(x+4) = \frac{1}{2}(1-x)$
2. $\frac{2}{5}(y-3) = \frac{1}{3}(y+2)$
3. $\frac{3}{8}(2z+1) = \frac{1}{4}(z-5)$

Remember to follow the same steps we used in the example: clear fractions, distribute, combine like terms, isolate the variable, and check your solution. Solving these equations will solidify your understanding and boost your confidence in tackling algebraic problems. Happy solving, guys!