Ranuil'solvity
| Steps | Resulting equation |
| :---- | :----------------- |
| 1 | [tex]$2 x=\frac{1}{4}-\frac{3}{5} x=\frac{35}{4}$[/tex] |
| 2 | [tex]$\frac{7}{5} x-\frac{1}{4}=\frac{55}{4}$[/tex] |
| 3 | [tex]$\frac{7}{5} x=\frac{88}{4}$[/tex] |
| 4 | [tex]$x=10$[/tex] |
Step 1
Step 2
Step 3
Step 4
Answer (1)
Step 1 incorrectly combines two equations.
Separate the equation into two: 2 x = 4 1 − 5 3 x and 4 1 − 5 3 x = 4 35 .
Solve the first equation to get x = 52 5 .
Solve the second equation to get x = − 6 85 .
Explanation
Analyzing the Problem Let's analyze Ranuil's steps to identify the mistake in solving the equation. The goal is to pinpoint where the error occurred and understand why the final answer is incorrect.
Identifying the Error in Step 1 Step 1 combines two separate expressions into one equation: 2 x = 4 1 − 5 3 x = 4 35 . This is where the initial error lies. The equation should be split into two separate equations: 2 x = 4 1 − 5 3 x and 4 1 − 5 3 x = 4 35 .
Solving the First Equation Let's solve the first equation correctly: 2 x = 4 1 − 5 3 x . To do this, we add 5 3 x to both sides of the equation, which gives us 2 x + 5 3 x = 4 1 . Combining the terms on the left side, we get 5 10 x + 5 3 x = 5 13 x = 4 1 .
Solving the Second Equation Now, let's solve the second equation correctly: 4 1 − 5 3 x = 4 35 . Subtract 4 1 from both sides: − 5 3 x = 4 35 − 4 1 = 4 34 = 2 17 . Multiply both sides by − 3 5 : x = 2 17 ⋅ − 3 5 = − 6 85 .
Identifying Errors in Subsequent Steps Since Step 1 is incorrect, all subsequent steps are also incorrect. Step 2, Step 3, and Step 4 are based on the flawed equation from Step 1. Therefore, the final answer x = 10 is wrong.
Correcting the Approach and Finding the Solutions The correct approach involves recognizing that Step 1 incorrectly combines two equations. Solving the two separate equations derived from Step 1 gives us x = 52 5 from the first equation ( 2 x = 4 1 − 5 3 x ) and x = − 6 85 from the second equation ( 4 1 − 5 3 x = 4 35 ). Since the original equation is flawed, there is no single correct solution for x .
Examples
When solving complex problems, like those in engineering or physics, it's crucial to break them down into smaller, manageable steps. Just as we identified the error in the initial step of Ranuil's solution, engineers must meticulously verify each stage of their calculations to avoid compounding errors. For instance, in designing a bridge, an initial miscalculation of load distribution could lead to catastrophic failure. Therefore, a step-by-step verification process is essential to ensure accuracy and reliability.
