Solving Equations by Interchanging Numbers and Signs

The question asks us to evaluate two equations after interchanging specific numbers and signs. We need to swap the number 7 with the number 9, and the multiplication sign (×) with the division sign (÷).

Let's apply these interchanges to each equation one by one and then calculate the resulting value using the order of operations (BODMAS/PEMDAS).

Applying Interchanges to Equation I

The original Equation I is: 8 × 3 ÷ 6 + 9 – 7

Applying the interchanges:

• Replace × with ÷: 8 ÷ 3
• Replace ÷ with ×: 8 ÷ 3 × 6
• The + and – signs remain the same.
• Replace 9 with 7: ... + 7
• Replace 7 with 9: ... – 9

The modified Equation I is: 8 ÷ 3 × 6 + 7 – 9

Calculating Equation I

Now, we calculate the value of the modified equation using the order of operations (BODMAS/PEMDAS - Brackets, Orders (powers/roots), Division/Multiplication (left-to-right), Addition/Subtraction (left-to-right)).

• Division/Multiplication (left to right):
• First, $8 \div 3$: This gives $\frac{8}{3}$.
• Next, $\frac{8}{3} \times 6$: $\frac{8}{3} \times 6 = \frac{8 \times 6}{3} = \frac{48}{3} = 16$.
• The equation becomes: 16 + 7 – 9
• Addition/Subtraction (left to right):
• First, $16 + 7 = 23$.
• Next, $23 – 9 = 14$.

The value of Equation I after interchanges is 14.

Applying Interchanges to Equation II

The original Equation II is: 9 – 7 × 1 + 6 ÷ 3

Applying the interchanges:

• Replace 9 with 7: 7 – ...
• Replace 7 with 9: ... 9 × ...
• The – and + signs remain the same.
• Replace × with ÷: ... – 9 ÷ 1 ...
• Replace ÷ with ×: ... + 6 × 3

The modified Equation II is: 7 – 9 ÷ 1 + 6 × 3

Calculating Equation II

Again, we use the order of operations (BODMAS/PEMDAS).

• Division/Multiplication (left to right):
• First, $9 \div 1 = 9$.
• Next, $6 \times 3 = 18$.
• The equation becomes: 7 – 9 + 18
• Addition/Subtraction (left to right):
• First, $7 – 9 = -2$.
• Next, $-2 + 18 = 16$.

The value of Equation II after interchanges is 16.

So, after interchanging the numbers and signs as specified, the value of equation (I) is 14 and the value of equation (II) is 16. The values respectively are 14, 16.

Revision Table: Key Concepts

Additional Information: BODMAS/PEMDAS

Understanding the correct order of operations is fundamental for evaluating mathematical expressions accurately. Let's break down BODMAS/PEMDAS:

• B/P: Brackets / Parentheses - Evaluate expressions inside brackets first.
• O/E: Orders / Exponents - Calculate powers, roots, or other orders next.
• DM: Division and Multiplication - Perform division and multiplication from left to right as they appear in the expression. They have equal priority.
• AS: Addition and Subtraction - Perform addition and subtraction from left to right as they appear in the expression. They have equal priority.

In the examples above, once we performed the division and multiplication from left to right, we then proceeded with addition and subtraction from left to right to get the final values.