Equation Error: Find The Mistake & Correct Solution

Hey there, math enthusiasts! Let's dive into a common scenario where a friend has tackled an equation but arrived at an incorrect solution. In this case, our friend solved the equation 4x + 24x - 2 = 7(4x + 2) and concluded that x = 16. But is that right? Let's put on our detective hats and figure out where things went wrong and, more importantly, how to find the correct solution.

Spotting the Error: A Step-by-Step Investigation

In mathematics, just like in detective work, the key to solving a problem lies in meticulous attention to detail. To pinpoint the error, we'll need to carefully retrace the steps our friend might have taken. It's likely that the mistake occurred during the simplification or solving process. Let's break down the equation and solve it step-by-step ourselves, comparing our work with the likely steps taken that led to the incorrect answer.

First, we rewrite the original equation for clarity:

4x + 24x - 2 = 7(4x + 2)

The first step in solving most equations is to simplify both sides. On the left side, we can combine the 'x' terms:

(Combining Like Terms): Combining like terms is a fundamental algebraic operation that simplifies equations by grouping terms with the same variable and exponent. In this case, both 4x and 24x are terms with 'x' to the power of 1, making them like terms. This step involves adding the coefficients of the like terms while keeping the variable the same, akin to adding apples to apples.

4x + 24x = 28x

So, the left side of the equation simplifies to:

28x - 2

On the right side, we need to distribute the 7 across the terms inside the parentheses:

(Distribution Property): The distributive property is a key concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. It's a way of breaking down a complex multiplication into simpler parts. Think of it as sharing: the number outside the parentheses gets shared with every number inside. Each term inside the parentheses is multiplied by the term outside, which involves both the numerical coefficient and the variable, if any. In simpler terms, it involves multiplying the number outside the parentheses with each term inside the parentheses, ensuring that every term within the grouping is properly accounted for in the multiplication process.

7 * 4x = 28x

7 * 2 = 14

Therefore, the right side of the equation becomes:

28x + 14

Now, our equation looks like this:

28x - 2 = 28x + 14

Now, let’s isolate the 'x' terms. To do that, we can subtract 28x from both sides of the equation. This is done to group all terms containing 'x' on one side and constant terms on the other, a crucial step in solving algebraic equations.

(Isolating Variables): Isolating variables is a central technique in solving equations, aimed at getting the variable by itself on one side of the equals sign. This often involves performing the same operation on both sides of the equation to maintain balance, like adding or subtracting a term from both sides. Think of it as a tug-of-war where you want to get the 'x' term alone on one side by strategically moving other terms around. This helps in simplifying the equation and revealing the value of the unknown variable. It’s a fundamental skill in algebra, turning complex equations into solvable forms.

28x - 28x - 2 = 28x - 28x + 14

This simplifies to:

-2 = 14

Wait a minute! This is a peculiar situation. We've arrived at a statement that is clearly false: -2 does not equal 14. This indicates that something interesting is happening with our equation. The equation that results in a false statement like this reveals that there's no value for 'x' that can make the original equation true.

(No Solution): In mathematics, equations that simplify to a false statement, like -2 = 14, have no solution. This means there is no value for the variable 'x' that can make the original equation true. It's a critical concept in algebra, highlighting that not every equation has a solvable answer. Recognizing when an equation has no solution is just as important as solving one, as it guides mathematical understanding and problem-solving approaches.

The Correct Solution: Understanding "No Solution"

So, what's the correct solution? The answer is that there is no solution to this equation. This is a crucial concept in algebra. Not every equation has a solution where x equals a specific number. Sometimes, the equation itself is contradictory.

Possible Error: A Misinterpretation

Our friend arrived at x = 16, which is incorrect. The most likely error is a misinterpretation of the algebraic steps or an arithmetic mistake during the simplification process. It's possible they made an error when distributing, combining like terms, or isolating the variable. Without seeing their exact steps, it's difficult to pinpoint the precise error, but the key takeaway is that careful attention to each step is crucial.

Key Takeaways for Avoiding Errors

1. Double-Check Your Work: Always review each step to ensure accuracy.
2. Simplify Carefully: Pay close attention to signs (positive and negative) and the order of operations.
3. Distribute Correctly: Make sure to multiply the term outside the parentheses by every term inside.
4. Combine Like Terms Methodically: Group similar terms together to simplify the equation.
5. Recognize "No Solution" Cases: Be aware that some equations do not have a numerical solution.

The Importance of Precision in Math

This exercise highlights the significance of precision in mathematics. A small error in one step can lead to a completely incorrect solution. By understanding the underlying principles and practicing careful problem-solving techniques, we can minimize mistakes and build confidence in our mathematical abilities.

So, next time you're tackling an equation, remember to take your time, double-check your work, and don't be afraid to ask for help if you get stuck. Math is a journey of exploration and discovery, and every mistake is an opportunity to learn and grow!

In conclusion, while our friend might have made a mistake leading to the incorrect solution x=16, the actual outcome of the equation 4x + 24x - 2 = 7(4x + 2) reveals that there is no solution. This understanding underscores the critical aspect of precision in mathematical problem-solving and the importance of acknowledging that not all equations yield a straightforward numerical answer.

For further learning and clarification on solving algebraic equations, consider exploring resources such as Khan Academy's Algebra Section.