Solving The Equation: (1-x) + 3/4 = 5/6

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Hey guys! Today, we're diving into a bit of algebra to solve the equation (1-x) + 3/4 = 5/6. It might look a little intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. We'll go through each part of the equation, explain the steps we're taking, and by the end, you'll not only know how to solve this specific problem but also have a better grasp on how to tackle similar algebraic equations. So, let's get started and make math a little less scary and a lot more fun!

Understanding the Basics of Algebraic Equations

Before we jump right into solving our equation, let's quickly refresh some algebraic fundamentals. Think of an equation like a balanced scale. On one side, you have an expression (in our case, (1-x) + 3/4), and on the other side, you have another expression or a value (in our case, 5/6). The goal is to find the value of the unknown variable (which is 'x' in our equation) that keeps the scale balanced. This means that whatever we do to one side of the equation, we must do to the other side to maintain that balance.

The basic operations we'll use are addition, subtraction, multiplication, and division. Remember, the key is to isolate the variable 'x' on one side of the equation. This often involves performing inverse operations. For example, if we see addition, we might need to use subtraction to 'undo' it. If we see multiplication, we might use division. Understanding these basic principles is crucial, as they form the foundation for solving more complex equations later on. So, let's keep these in mind as we move forward with solving our problem.

Step-by-Step Solution

Now, let's get into the nitty-gritty and solve the equation (1-x) + 3/4 = 5/6 step by step.

  1. Combine Constants: Our first goal is to simplify the left side of the equation by combining the constants, which are the numbers without any variables attached. We have '1' and '3/4'. To add these, we need a common denominator. We can rewrite '1' as '4/4'. So, we have 4/4 + 3/4, which equals 7/4. Our equation now looks like this: (7/4) - x = 5/6.

  2. Isolate the Variable Term: Next, we want to isolate the term with 'x' on one side. Currently, we have '-x' which we want to isolate. To do this, we need to get rid of the '7/4' that's being added (or, in this case, we're subtracting x from it). We do this by subtracting 7/4 from both sides of the equation. This gives us: -x = 5/6 - 7/4.

  3. Find a Common Denominator: To subtract 7/4 from 5/6, we need a common denominator. The least common multiple of 6 and 4 is 12. So, we'll convert both fractions to have a denominator of 12. 5/6 becomes 10/12 (multiply both numerator and denominator by 2), and 7/4 becomes 21/12 (multiply both numerator and denominator by 3). Now our equation looks like: -x = 10/12 - 21/12.

  4. Subtract the Fractions: Now we can subtract the fractions on the right side: 10/12 - 21/12 = -11/12. So, we have -x = -11/12.

  5. Solve for x: We have '-x', but we want 'x'. To get 'x', we can multiply both sides of the equation by -1. This changes the signs of both sides. So, -x becomes x, and -11/12 becomes 11/12. Therefore, x = 11/12.

So, after walking through each step, we've found that the solution to the equation (1-x) + 3/4 = 5/6 is x = 11/12. It’s all about breaking down the problem into manageable steps and remembering to keep the equation balanced!

Common Mistakes to Avoid

When solving equations like (1-x) + 3/4 = 5/6, it’s super easy to make a few common mistakes. Knowing these pitfalls can save you a lot of headaches and help you get to the correct answer more efficiently. Let's chat about some of the frequent slip-ups.

  1. Forgetting the Order of Operations: One of the biggest culprits is forgetting the order of operations (PEMDAS/BODMAS). This acronym reminds us of the order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our equation, we need to deal with the parentheses first, but sometimes people might jump to adding fractions before simplifying inside the parentheses, which can lead to errors.

  2. Incorrectly Combining Fractions: Fractions can be tricky! A common mistake is trying to add or subtract fractions without finding a common denominator first. Remember, you can only add or subtract fractions if they have the same denominator. In our problem, we needed to convert 3/4 and 5/6 to fractions with a common denominator of 12 before we could perform any operations.

  3. Sign Errors: Sign errors are super common, especially when dealing with negative numbers. It’s essential to pay close attention to the signs of the numbers and variables throughout the equation. For instance, when we had -x = -11/12, we needed to remember to multiply both sides by -1 to get x = 11/12. Forgetting this last step, or making a mistake with the signs, can lead to the wrong answer.

  4. Not Keeping the Equation Balanced: As we discussed earlier, an equation is like a balanced scale. Whatever operation you perform on one side, you must perform on the other side to maintain the balance. A common mistake is to only apply an operation to one side, which throws off the entire equation and leads to an incorrect solution.

  5. Rushing Through the Steps: Math problems, especially algebraic equations, require careful attention to detail. Rushing through the steps can lead to silly mistakes, like copying a number incorrectly or skipping a necessary operation. It's always a good idea to take your time, double-check your work, and make sure each step is accurate.

By being mindful of these common mistakes, you can approach algebraic equations with more confidence and accuracy. Remember, practice makes perfect, so the more you solve these types of problems, the better you'll become at avoiding these pitfalls.

Alternative Methods to Solve the Equation

Okay, so we've tackled the equation (1-x) + 3/4 = 5/6 using a step-by-step method. But guess what? There's often more than one way to skin a cat, and the same goes for solving algebraic equations! Let's explore some alternative methods that can give you a different perspective and help you become an even more versatile problem-solver. These methods can sometimes be quicker or more intuitive, depending on your personal preference and the specific equation you're dealing with.

  1. Clearing Fractions from the Start: Instead of dealing with fractions throughout the problem, you can eliminate them right off the bat. This involves finding the least common multiple (LCM) of the denominators and multiplying both sides of the equation by that LCM. In our case, the denominators are 4 and 6, and their LCM is 12. So, we can multiply both sides of the equation by 12. This gives us: 12 * [(1-x) + 3/4] = 12 * (5/6). Distribute the 12 on the left side: 12 * (1-x) + 12 * (3/4) = 12 * (5/6). This simplifies to: 12 - 12x + 9 = 10. Now you have an equation without fractions, which some people find easier to work with. From here, you can combine like terms and isolate 'x' as we did in the original method.

  2. Rearranging the Equation First: Another approach is to rearrange the equation before combining constants. For instance, you could start by adding 'x' to both sides of the equation. This gives you: 1 + 3/4 = 5/6 + x. Then, you can subtract 5/6 from both sides to isolate 'x'. This method might be helpful if you prefer to keep 'x' positive from the beginning. The steps would look like this: 1 + 3/4 - 5/6 = x. Now you just need to find a common denominator and combine the fractions to find the value of 'x'.

  3. Using a Visual Model: For some people, visualizing the equation can make it easier to understand and solve. You could represent the equation using a bar model or a diagram. For example, you could draw a bar to represent '1', another to represent '3/4', and another to represent '5/6'. Then, you can visually manipulate the bars to isolate 'x'. This method is especially helpful for those who are visual learners or who are just starting to learn algebra.

Each of these alternative methods offers a slightly different way to approach the problem, and each has its own advantages. The best method for you will depend on your individual style and the specific problem you're trying to solve. The key is to be flexible and willing to try different approaches until you find one that clicks for you. Remember, math is all about problem-solving, and there's often more than one path to the solution!

Real-World Applications of Solving Equations

So, we've nailed how to solve the equation (1-x) + 3/4 = 5/6, which is awesome! But you might be wondering,