Solving Equations by Interchanging Signs and Numbers

This problem requires us to evaluate two equations after applying specific rules for interchanging signs and numbers. We need to swap the multiplication sign (×) with the division sign (÷) and the number 2 with the number 6 in both equations. After performing the interchanges, we will check if the resulting equations are correct based on standard mathematical operations.

Rules for Interchange

• × will be replaced by ÷
• ÷ will be replaced by ×
• The number 2 will be replaced by the number 6
• The number 6 will be replaced by the number 2

Analysis of Equation I

The original Equation I is:

\(7 - 4 \times 3 \div 6 + 2 = 8\)

Now, let's apply the given interchanges to Equation I:

• Replace × with ÷
• Replace ÷ with ×
• Replace 6 with 2
• Replace 2 with 6

Applying these rules, the new equation becomes:

\(7 - 4 \div 3 \times 2 + 6\)

Let's evaluate this expression using the order of operations (BODMAS/PEMDAS):

1. Perform division and multiplication from left to right:
2. \(4 \div 3 = \frac{4}{3}\)
3. Then, \(\frac{4}{3} \times 2 = \frac{8}{3}\)
4. Perform addition and subtraction from left to right:
5. \(7 - \frac{8}{3} + 6\)
6. Combine terms: \(7 + 6 - \frac{8}{3} = 13 - \frac{8}{3}\)
7. Find a common denominator (3): \(13 = \frac{13 \times 3}{3} = \frac{39}{3}\)
8. Subtract: \(\frac{39}{3} - \frac{8}{3} = \frac{39 - 8}{3} = \frac{31}{3}\)

The calculated value of the left side is \(\frac{31}{3}\). The original equation stated the result is 8. Since \(\frac{31}{3} \neq 8\) (because \(8 = \frac{24}{3}\)), Equation I is NOT correct after the interchange.

Analysis of Equation II

The original Equation II is:

\(6 - 8 \times 2 + 9 \div 3 = 5\)

Now, let's apply the given interchanges to Equation II:

• Replace × with ÷
• Replace ÷ with ×
• Replace 6 with 2
• Replace 2 with 6

Applying these rules, the new equation becomes:

\(2 - 8 \div 6 + 9 \times 3\)

Let's evaluate this expression using the order of operations (BODMAS/PEMDAS):

1. Perform division and multiplication from left to right:
2. \(8 \div 6 = \frac{8}{6} = \frac{4}{3}\)
3. \(9 \times 3 = 27\)
4. Perform addition and subtraction from left to right:
5. \(2 - \frac{4}{3} + 27\)
6. Combine terms: \(2 + 27 - \frac{4}{3} = 29 - \frac{4}{3}\)
7. Find a common denominator (3): \(29 = \frac{29 \times 3}{3} = \frac{87}{3}\)
8. Subtract: \(\frac{87}{3} - \frac{4}{3} = \frac{87 - 4}{3} = \frac{83}{3}\)

The calculated value of the left side is \(\frac{83}{3}\). The original equation stated the result is 5. Since \(\frac{83}{3} \neq 5\) (because \(5 = \frac{15}{3}\)), Equation II is NOT correct after the interchange.

Conclusion

After interchanging the signs (× and ÷) and the numbers (2 and 6):

• Equation I results in \(\frac{31}{3}\), which is not equal to 8. So, Equation I is not correct.
• Equation II results in \(\frac{83}{3}\), which is not equal to 5. So, Equation II is not correct.

Therefore, both Equation I and Equation II will be not correct after the specified interchanges.

Revision Table - Interchanging Operators and Numbers

Additional Information - Order of Operations (BODMAS/PEMDAS)

When evaluating mathematical expressions, we follow a specific order of operations to ensure consistency in results. This order is often remembered using acronyms like BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• PEMDAS: Parentheses, Exponents (powers/roots), Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In this problem, we used Division and Multiplication before Addition and Subtraction, working from left to right for operations at the same level.