Solving The Equation (4/3)x=1/9 A Step-by-Step Guide

Hey guys! Ever find yourself staring at an equation, feeling like you're trying to decipher ancient hieroglyphics? Well, you're not alone! Equations can seem intimidating, but with a little understanding and a systematic approach, you can conquer them. In this guide, we're going to break down the process of solving equations, focusing on justifying each step along the way. We'll use a specific example to illustrate the concepts, but the principles apply to a wide range of equations. So, buckle up, and let's dive into the world of equation solving!

Understanding the Basics of Equations

Before we jump into the nitty-gritty, let's make sure we're all on the same page with some fundamental concepts. An equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale – both sides must have the same weight for the scale to remain level. Our goal in solving an equation is to isolate the variable (usually represented by a letter like x or y) on one side of the equation. This means we want to get the variable by itself, with a coefficient of 1. To do this, we use inverse operations, which are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division. The key principle we'll be using throughout this process is the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side to maintain the balance.

The Multiplication Property of Equality

The multiplication property of equality is a cornerstone of solving equations. It states that if you multiply both sides of an equation by the same non-zero number, the equation remains true. This property is crucial for eliminating fractions and isolating variables. Let's say we have an equation like x/2 = 5. To get x by itself, we can multiply both sides of the equation by 2. This gives us (2) * (x/2) = (2) * 5, which simplifies to x = 10. See how multiplying both sides by 2 allowed us to isolate x? We'll be using this property extensively in our example.

The Division Property of Equality

Similar to the multiplication property, the division property of equality states that if you divide both sides of an equation by the same non-zero number, the equation remains true. This is another powerful tool for isolating variables. For instance, if we have the equation 3x = 12, we can divide both sides by 3 to get x = 4. Both the multiplication and division properties are based on the fundamental principle of maintaining balance in the equation. It's like making sure both sides of our scale remain equal by performing the same operation on each side. When we solve equations, we're essentially unwrapping the variable to expose its true value.

Solving the Equation Step-by-Step

Okay, let's tackle the equation you provided:

(4/3) * x = 1/9

This equation involves a fraction multiplied by our variable x. Our goal is to isolate x, which means we need to get rid of the fraction (4/3). How do we do that? This is where the concept of a reciprocal comes into play. The reciprocal of a fraction is simply that fraction flipped over. So, the reciprocal of 4/3 is 3/4. When you multiply a fraction by its reciprocal, you get 1. This is incredibly useful for eliminating fractions in equations. Let's walk through the solution step-by-step, justifying each step as we go.

Step 1 Multiplying by the Reciprocal

To isolate x, we need to get rid of the 4/3 that's multiplying it. As we discussed, we can do this by multiplying both sides of the equation by the reciprocal of 4/3, which is 3/4. This gives us:

(3/4) * (4/3) * x = (3/4) * (1/9)

Justification: We're using the multiplication property of equality here. We're multiplying both sides of the equation by the same value (3/4), which ensures that the equation remains balanced. This is a crucial step in isolating the variable x.

Step 2 Simplifying the Left Side

Now, let's simplify the left side of the equation. We have (3/4) * (4/3) * x. Remember that when you multiply fractions, you multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (3/4) * (4/3) is (3 * 4) / (4 * 3), which equals 12/12. And 12/12 is simply 1. So, the left side of the equation simplifies to 1 * x, which is just x. This simplifies our equation to:

x = (3/4) * (1/9)

Justification: We're using the multiplicative inverse property here. This property states that any number multiplied by its reciprocal equals 1. By multiplying 4/3 by its reciprocal 3/4, we effectively eliminate the fraction and get 1, leaving us with x by itself. This is exactly what we wanted to achieve.

Step 3 Simplifying the Right Side

Now, let's simplify the right side of the equation, which is (3/4) * (1/9). Again, we multiply the numerators and the denominators: (3 * 1) / (4 * 9) = 3/36. Now, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 3. This gives us:

x = 1/12

Justification: We're using the properties of fraction multiplication and simplification of fractions. Multiplying the fractions involves multiplying numerators and denominators. Simplifying the resulting fraction involves dividing both the numerator and denominator by their greatest common factor, which doesn't change the value of the fraction, just its appearance.

The Complete Solution and Justification Summary

So, the complete solution to the equation (4/3) * x = 1/9 is:

x = 1/12

Let's recap the steps and their justifications:

1. (3/4) * (4/3) * x = (3/4) * (1/9) - Justification: Multiplication property of equality. Multiplying both sides by the reciprocal of 4/3 (which is 3/4) to isolate x.
2. x = (3/4) * (1/9) - Justification: Multiplicative inverse property. 4/3 multiplied by its reciprocal 3/4 equals 1.
3. x = 1/12 - Justification: Properties of fraction multiplication and simplification of fractions. Multiplying the fractions and simplifying the result.

Why Justification is Key

Guys, justifying each step in solving an equation isn't just about following rules – it's about understanding the why behind those rules. When you understand the underlying principles, you're not just memorizing steps; you're developing a deeper understanding of mathematics. This deeper understanding will help you tackle more complex problems and apply these concepts in different contexts. Plus, justifying your steps helps you catch errors and ensures that your solution is logically sound.

Practice Makes Perfect

Solving equations is a skill that gets better with practice. The more equations you solve, the more comfortable you'll become with the process. So, don't be afraid to tackle different types of equations, and always remember to justify each step. If you get stuck, review the properties and principles we discussed, and break the problem down into smaller, manageable steps. You've got this!

Conclusion

Solving equations might seem daunting at first, but by understanding the underlying principles and justifying each step, you can approach them with confidence. Remember the golden rule of algebra, the properties of equality, and the power of inverse operations. Keep practicing, and you'll become a master equation solver in no time! So go ahead guys, keep solving, keep learning, and keep justifying!