Malik's Mistake In Solving A Linear Equation A Detailed Solution

Malik attempted to solve the equation $\frac{2}{5}x - 4y = 10$ for $y$ when $x = 60$. However, there's a critical error in his approach. Let's break down Malik's steps, identify the mistake, and correctly solve the problem.

Malik's Attempted Solution

Malik's work is presented as follows:

$\begin{array}{l} \frac{2}{5} x-4 y=10 \\ \frac{2}{5} x-4(60)=10 \\ \frac{2}{5} x-240=10 \\ \frac{2}{5} x-240+240=10+240 \\ \frac{5}{2}\left[\frac{2}{5} x=\frac{5}{2}(250)\right] \\ x=625 \end{array}$

Identifying the Error

The fundamental error lies in the second step. Malik incorrectly substitutes $x=60$ into the equation. He seems to be replacing the y variable instead of the x. This initial mistake propagates through the rest of the solution, leading to a wrong answer. Malik should have substituted 60 for x in the original equation, which would then allow him to solve for the correct value of y. Let’s clarify why this is a significant error. In algebraic equations, each variable represents an unknown quantity. When we are given a specific value for a variable, such as x = 60, it means we are told exactly what number that variable represents in the context of the problem. Substituting this value correctly is essential to find the corresponding value of other variables that satisfy the equation. The incorrect substitution completely changes the relationship described by the equation, leading to an incorrect solution. This highlights the importance of carefully reading and understanding the problem statement before attempting to solve it. Accurate substitution is a basic yet critical skill in algebra, and errors at this stage can nullify all subsequent steps.

Correct Solution: Step-by-Step

To correctly solve for y when x = 60, we need to follow these steps:

1. Substitute the value of x into the original equation: $\frac{2}{5}(60) - 4y = 10$

2. Simplify the equation by performing the multiplication: $24 - 4y = 10$

3. Isolate the term with y by subtracting 24 from both sides: $24 - 4y - 24 = 10 - 24$ $-4y = -14$

4. Solve for y by dividing both sides by -4: $\frac{-4y}{-4} = \frac{-14}{-4}$ $y = \frac{7}{2}$ or $y = 3.5$

Therefore, the correct solution is $y = 3.5$ when $x = 60$. This correct solution provides a clear contrast to Malik’s incorrect answer, emphasizing the importance of accurate substitution and algebraic manipulation. The steps taken in the correct solution are standard algebraic procedures used to isolate and solve for a variable. By substituting x = 60, we transform the equation into one where y is the only unknown. We then use inverse operations (subtraction and division) to isolate y on one side of the equation, thus finding its value. Each step is crucial, and performing them in the correct order ensures that the solution is accurate. This process not only solves the problem but also reinforces fundamental algebraic skills that are essential for more complex mathematical problems.

Detailed Explanation of Each Correct Step

Let's delve deeper into each step of the correct solution to understand the underlying mathematical principles:

• Step 1: Substitution

  The initial equation is $\frac{2}{5}x - 4y = 10$. When we substitute $x = 60$, we replace the x in the equation with its given value. This gives us $\frac{2}{5}(60) - 4y = 10$. Substitution is a fundamental concept in algebra, allowing us to evaluate expressions and solve equations by replacing variables with specific values. In this context, it helps us transform a two-variable equation into a single-variable equation, which is easier to solve. The accuracy of the substitution is paramount; any mistake here will propagate through the rest of the solution. It’s crucial to ensure that the correct value is placed in the correct location within the equation. This step lays the groundwork for the subsequent steps, which build upon this initial transformation.

• Step 2: Simplification

  We simplify the equation by performing the multiplication: $\frac{2}{5}(60) = 24$. The equation then becomes $24 - 4y = 10$. Simplification involves reducing the equation to its simplest form by performing arithmetic operations. Here, we multiply the fraction by the integer, making the equation more manageable. Simplifying complex terms early in the process can prevent errors and make the subsequent steps easier to execute. This step bridges the gap between the initial substitution and the isolation of the variable, streamlining the process towards the final solution. The simplified equation now clearly shows the relationship between the remaining terms and the variable y, setting the stage for the next steps.

• Step 3: Isolation of the Variable Term

  To isolate the term with y, we subtract 24 from both sides of the equation: $24 - 4y - 24 = 10 - 24$, which simplifies to $-4y = -14$. Isolating the variable term is a crucial algebraic technique used to separate the variable we want to solve for from other terms in the equation. By performing the same operation on both sides, we maintain the equation's balance while moving closer to isolating y. Subtraction is used here as the inverse operation of addition, effectively canceling out the constant term on the left side of the equation. This step transforms the equation into a simpler form where only the variable term and a constant are present, making it straightforward to solve for y in the next step.

• Step 4: Solving for y

  Finally, we solve for y by dividing both sides of the equation by -4: $\frac{-4y}{-4} = \frac{-14}{-4}$, which gives us $y = \frac{7}{2}$ or $y = 3.5$. Solving for the variable involves performing the inverse operation to isolate the variable completely. Here, we divide by -4, which is the coefficient of y, to find the value of y. Division is the inverse operation of multiplication, and it undoes the multiplication of -4 with y. This final step provides the numerical value of y that satisfies the equation when x is 60. The result, $y = 3.5$, is the solution to the problem and represents the value of y that makes the equation true under the given condition.

The Importance of Checking the Solution

To ensure the solution is correct, we can substitute both $x = 60$ and $y = 3.5$ back into the original equation:

$\frac{2}{5}(60) - 4(3.5) = 10$

$24 - 14 = 10$

$10 = 10$

Since the equation holds true, our solution is correct. Checking the solution is a vital step in problem-solving. It confirms that the calculated values satisfy the original equation, reducing the likelihood of errors. This process involves substituting the found values back into the equation and verifying that both sides are equal. If the equation does not hold true, it indicates a mistake in the solution process, prompting a review of the steps taken. Checking provides a level of confidence in the correctness of the answer and reinforces the understanding of the equation's properties. It's a good practice to incorporate this step into every problem-solving routine.

Common Mistakes in Solving Linear Equations

Malik's mistake highlights a few common pitfalls when solving linear equations. Understanding these common errors can help prevent them in the future:

• Incorrect Substitution: As seen in Malik's attempt, substituting a value for the wrong variable is a frequent mistake. Always double-check which variable's value is given and ensure it's correctly placed in the equation. Incorrect substitution leads to a completely different equation, rendering the subsequent steps meaningless. It’s crucial to pay close attention to the problem statement and understand exactly which variable is assigned a value. Taking the time to verify the substitution can prevent significant errors later on.

• Arithmetic Errors: Simple arithmetic mistakes, such as incorrect multiplication or division, can lead to wrong answers. Double-check each calculation to avoid these errors. Arithmetic errors are subtle but can have a significant impact on the solution. Careful attention to detail and a systematic approach can help minimize these mistakes. Using a calculator or performing calculations twice can also help ensure accuracy.

• Incorrect Order of Operations: Failing to follow the order of operations (PEMDAS/BODMAS) can result in errors. Remember to perform multiplication and division before addition and subtraction. The order of operations is a fundamental principle in mathematics that dictates the sequence in which operations should be performed. Ignoring this order can lead to incorrect simplification and ultimately an incorrect solution. Brackets or parentheses should be dealt with first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

• Not Performing the Same Operation on Both Sides: When manipulating an equation, it's crucial to perform the same operation on both sides to maintain equality. Forgetting this rule can unbalance the equation and lead to an incorrect solution. Maintaining balance in an equation is critical to solving it correctly. Any operation performed on one side must be mirrored on the other side to preserve the equality. This principle is the foundation of algebraic manipulation and ensures that the solution remains valid.

Conclusion

Malik's attempt to solve the equation $\frac{2}{5}x - 4y = 10$ when $x = 60$ serves as a valuable lesson in the importance of careful substitution and accurate algebraic manipulation. By identifying the error, working through the correct solution, and understanding common mistakes, we can improve our problem-solving skills and avoid similar errors in the future. The correct solution to the equation $\frac{2}{5}x - 4y = 10$ when $x = 60$ is $y = 3.5$. This detailed analysis underscores the significance of each step in solving linear equations and the need for precision in algebraic manipulations.