Solve X * 56/11 - 12 * 23 * X ÷ (2/3) = 1/2
Introduction
Hey guys! Today, we're diving into a cool math problem that involves solving for x in a somewhat complex equation. Don't worry, it's not as scary as it looks! We're going to break it down step-by-step so it's super easy to follow. Our main goal here is to isolate x on one side of the equation. Remember, the key to solving any equation is to keep it balanced – what you do on one side, you gotta do on the other. So, let’s jump right in and see how we can tackle this! We'll start by simplifying the equation, then we'll move terms around, and finally, we'll solve for x. Are you ready? Let's get started and make math fun!
Understanding the Equation
Okay, let’s first understand the equation we're dealing with: x * 56 / 11 - 12 * 1 * 23 * x ÷ (2/3) = 1/2. This might look a bit intimidating at first, but trust me, we can handle it. The equation involves a bunch of operations: multiplication, division, subtraction, and fractions. The most important thing to remember is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This tells us the order in which we should perform the operations. First, we'll deal with any parentheses, then exponents, then multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Before we start crunching numbers, let's rewrite the equation slightly to make it clearer. We can rewrite the division by a fraction as multiplication by its reciprocal. This will make it easier to handle. So, instead of dividing by 2/3, we'll multiply by 3/2. This little trick often simplifies things and makes the equation less confusing. Remember, math is all about finding the easiest way to solve a problem, and this is one of those tricks that can save us a lot of time and effort. Now, let’s move on to the next step and start simplifying!
Step-by-Step Solution
Step 1: Simplify the terms
First, let’s simplify the terms in the equation. We have x * 56 / 11, which we can leave as is for now. Then we have 12 * 1 * 23 * x, which simplifies to 276x. And we are dividing this by 2/3, which, as we discussed, is the same as multiplying by 3/2. So, the term becomes 276x * (3/2). Let's calculate that: 276 * (3/2) = 414. So, this part of the equation becomes 414x. Now, let's rewrite the whole equation with these simplifications: (56/11)x - 414x = 1/2. See? It's already looking less scary! We've taken a big chunk of the equation and made it more manageable. This is a crucial step because it helps us see the structure of the equation more clearly. Simplifying terms first makes the subsequent steps much easier. It's like clearing the clutter before you start organizing – you can see what you're working with much better. Next, we'll combine the x terms, but first, we need to find a common denominator to make the subtraction possible. So, let's move on to the next step and tackle that!
Step 2: Combine like terms
Okay, now we need to combine the x terms. We have (56/11)x - 414x. To subtract these, we need a common denominator. The easiest way to do this is to convert 414x into a fraction with a denominator of 11. So, we multiply 414 by 11, which gives us 4554. Therefore, 414x is the same as (4554/11)x. Now we can rewrite the equation as (56/11)x - (4554/11)x = 1/2. Next, we subtract the fractions: 56/11 - 4554/11 = -4498/11. So, the equation now looks like this: (-4498/11)x = 1/2. We're getting closer! Combining like terms is a fundamental step in solving equations. It allows us to consolidate the variable terms, making it easier to isolate x. Finding a common denominator might seem tedious, but it's essential for accurate subtraction or addition of fractions. This step has significantly simplified our equation, and now we're just one step away from finding the value of x. Next, we'll isolate x by multiplying both sides of the equation by the reciprocal of the coefficient of x. Let’s see how that works!
Step 3: Isolate x
Alright, we're in the home stretch! We have the equation (-4498/11)x = 1/2. To isolate x, we need to get rid of the fraction in front of it. We do this by multiplying both sides of the equation by the reciprocal of -4498/11, which is -11/4498. So, we multiply both sides by -11/4498: (-11/4498) * (-4498/11)x = (1/2) * (-11/4498). On the left side, the fractions cancel out, leaving us with just x. On the right side, we multiply the fractions: (1/2) * (-11/4498) = -11/8996. So, we have x = -11/8996. Now, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 11. So, -11/8996 simplifies to -1/818. Therefore, the solution to the equation is x = -1/818. Woohoo! We did it! Isolating x is the final step in solving for the variable. Multiplying by the reciprocal is a neat trick that allows us to cancel out the coefficient of x. Simplifying the fraction at the end gives us the solution in its simplest form. We've successfully navigated through the equation, step-by-step, and found the value of x. Pat yourselves on the back, guys! You're becoming math wizards!
Final Answer
So, after all that hard work, we've arrived at the final answer. The value of x that satisfies the equation x * 56 / 11 - 12 * 1 * 23 * x ÷ (2/3) = 1/2 is x = -1/818. Isn't it satisfying to solve a tricky problem like this? We started with a seemingly complicated equation, but by breaking it down into manageable steps, we were able to find the solution. Remember, guys, math is like a puzzle. Each step is a piece, and when you put them all together, you get the complete picture. We simplified the terms, combined like terms, and isolated x. Each step was crucial, and by following the order of operations and applying basic algebraic principles, we cracked the code. So, the next time you see a daunting equation, don't panic! Just take a deep breath, break it down, and tackle it step-by-step. You've got this! And now, you can confidently say that you know how to solve equations like this one. Great job, everyone! You're on your way to becoming math masters!