Solve -6/(a-4) = -8: Step-by-Step Guide

Hey guys! Let's dive into this math problem together and figure out how to solve for a in the equation -6/(a-4) = -8. This might look a bit intimidating at first, but don't worry, we'll break it down step by step and make it super clear. This article will help you master solving rational equations. So, grab your pencils, and let's get started!

Understanding the Equation

Before we jump into the solution, let's make sure we understand what the equation is telling us. We have a fraction, -6 divided by (a-4), and this whole expression is equal to -8. Our mission is to find the value of a that makes this equation true. Think of it like a puzzle – we need to find the missing piece that fits perfectly. In this case, the missing piece is the value of a. The equation we are tasked with solving is -6/(a-4) = -8. Understanding the components of this equation is crucial for finding the correct solution. Let's break it down:

• Fraction: On the left side, we have a fraction where -6 is the numerator (the top part) and (a-4) is the denominator (the bottom part). This fraction represents a division operation.
• Variable: The variable a is what we're trying to find. It's a placeholder for a number that, when plugged into the equation, will make the equation true.
• Equality: The equals sign (=) tells us that the expression on the left side of the equation has the same value as the expression on the right side. In this case, the fraction -6/(a-4) is equal to -8.
• Constant: On the right side, we have a constant, which is the number -8. Constants are values that don't change.

To solve for a, we need to isolate it on one side of the equation. This means we want to get a by itself, with no other numbers or operations attached to it. We'll do this by performing a series of algebraic operations that maintain the equality of the equation. Remember, whatever we do to one side of the equation, we must also do to the other side to keep things balanced. Understanding these basic components is the foundation for solving not just this equation, but many other algebraic problems as well. So, with a clear understanding of the equation's structure, we can move on to the next step: eliminating the fraction.

Step 1: Eliminating the Fraction

The first thing we want to do is get rid of that fraction. Fractions can sometimes make equations look more complicated than they really are. To eliminate the fraction, we're going to multiply both sides of the equation by the denominator, which is (a-4). This is a crucial step in isolating a. This technique is a common strategy in solving equations involving fractions. By multiplying both sides by the denominator, we effectively cancel out the fraction, making the equation easier to work with. It's like clearing away the clutter so we can see the core of the problem more clearly.

Here's how it looks:

-6/(a-4) = -8

Multiply both sides by (a-4):

(a-4) * [-6/(a-4)] = -8 * (a-4)

On the left side, (a-4) in the numerator and denominator cancel each other out, leaving us with just -6. On the right side, we have -8 multiplied by (a-4). So, after this step, our equation looks like this:

-6 = -8(a-4)

Now, we have a much simpler equation to work with. We've successfully eliminated the fraction, and we're one step closer to isolating a. Remember, the key to solving equations is to perform operations that maintain the balance – what we do to one side, we must do to the other. By multiplying both sides by (a-4), we've kept the equation balanced while simplifying its form. This step sets the stage for the next part of our solution, which involves distributing the -8 on the right side of the equation. By eliminating the fraction, we've transformed the equation into a more manageable form, making it easier to isolate and solve for a. This is a fundamental technique in algebra, and mastering it will help you tackle a wide range of equations.

Step 2: Distribute the -8

Now that we've gotten rid of the fraction, the next step is to distribute the -8 on the right side of the equation. This means we're going to multiply -8 by both terms inside the parentheses: a and -4. Distributing is like sharing; we're making sure the -8 interacts with each term individually. This step is essential because it helps us to further simplify the equation and get closer to isolating the variable a. Distributing properly ensures that we maintain the equality of the equation and accurately account for all terms. Let's see how it's done:

We have:

-6 = -8(a-4)

Distribute the -8:

-6 = (-8 * a) + (-8 * -4)

-6 = -8a + 32

So, after distributing the -8, our equation becomes -6 = -8a + 32. Notice how the -8 has been multiplied by both a and -4, resulting in -8a and +32, respectively. It's important to pay attention to the signs here – a negative times a negative gives a positive. Now, our equation looks even simpler. We've removed the parentheses, and we're ready to start isolating the term with a in it. Distributing is a key skill in algebra, and it's used in many different types of equations. By mastering this technique, you'll be well-equipped to handle more complex problems. The next step will involve moving the constant term (32) to the other side of the equation so we can get -8a by itself. Remember, our ultimate goal is to get a alone on one side, and each step we take brings us closer to that goal. Now that we've distributed the -8, the equation is in a form that's much easier to manipulate and solve.

Step 3: Isolate the Term with a

Alright, we're making great progress! We've eliminated the fraction and distributed the -8. Now, it's time to isolate the term with a in it, which is -8a. To do this, we need to get rid of the +32 that's on the same side of the equation. We can do this by subtracting 32 from both sides. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. This step is crucial because it brings us closer to having a all by itself. Isolating terms is a fundamental algebraic technique that allows us to simplify equations and solve for variables. By subtracting 32 from both sides, we're effectively moving the constant term away from the term containing a, which sets us up for the final step of dividing to solve for a. Let's see how it looks:

We have:

-6 = -8a + 32

Subtract 32 from both sides:

-6 - 32 = -8a + 32 - 32

-38 = -8a

So, after subtracting 32 from both sides, our equation becomes -38 = -8a. The +32 and -32 on the right side cancel each other out, leaving us with just -8a. Now, we have a much simpler equation with the term containing a isolated on one side. We're just one step away from solving for a! This step highlights the importance of maintaining balance in equations. By performing the same operation on both sides, we ensure that the equality remains true. Now that we've isolated -8a, the final step is to divide both sides by -8 to solve for a. This will give us the value of a that satisfies the original equation. By isolating the term with a, we've set the stage for the final calculation that will reveal the solution.

Step 4: Solve for a

We're in the home stretch now! We've done the hard work of eliminating the fraction, distributing, and isolating the term with a. Now, all that's left to do is solve for a by dividing both sides of the equation by -8. This will give us the value of a that makes the equation true. This step is the culmination of all our previous efforts, and it's where we finally get to see the solution. Dividing both sides by the coefficient of a is a standard technique in algebra, and it's essential for isolating the variable. By performing this division, we'll have a all by itself on one side of the equation, and the value on the other side will be our answer. Let's do it:

We have:

-38 = -8a

Divide both sides by -8:

-38 / -8 = -8a / -8

19/4 = a

So, after dividing both sides by -8, we find that a = 19/4. A negative divided by a negative gives a positive, so -38 / -8 simplifies to 19/4. We've done it! We've successfully solved for a. This final step demonstrates the power of algebraic manipulation. By carefully applying the rules of algebra, we can isolate variables and find solutions to equations. Now, we have the value of a that satisfies the original equation. It's always a good idea to check your answer by plugging it back into the original equation to make sure it works. In this case, if we substitute 19/4 for a in the original equation, we'll see that it does indeed make the equation true. This confirms that our solution is correct. The value of a that solves the equation is 19/4, which we have found by systematically simplifying the equation and isolating the variable.

Final Answer

So, the solution to the equation -6/(a-4) = -8 is a = 19/4. Great job, guys! We've walked through each step together, and now you know how to solve equations like this. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time!

One or more solutions: a = 19/4