Balancing Mathematical Equations by Substituting Signs

The question asks us to find the correct combination of mathematical signs to replace the '@' symbols in the given equation so that it balances. We are provided with a complex expression involving nested brackets and several '@' placeholders.

The equation structure is given as:

\({[\{(14 @ 6) @ (2 @ 3)\} @ (1 @ 7)] @ 2 @ 4}\)

Let's carefully identify the positions of the '@' symbols within the structure. There are a total of seven '@' symbols:

1. Between 14 and 6
2. Between the result of \((14 @ 6)\) and the result of \((2 @ 3)\)
3. Between 2 and 3
4. Between the result of \(\{(14 @ 6) @ (2 @ 3)\}\) and the result of \((1 @ 7)\)
5. Between 1 and 7
6. Between the result of the entire expression within the square brackets \([\dots]\) and the number 2
7. Between 2 and 4

The options provided contain a sequence of seven mathematical signs. The last sign in each option is '=', which indicates that the last '@' symbol represents the equality sign, and the number '4' is on the right side of the equation. Thus, the equation we need to balance is:

\({[\{(14 @ 6) @ (2 @ 3)\} @ (1 @ 7)] @ 2 = 4}\)

We need to test the given options by substituting the signs sequentially into the positions of the '@' symbols. Let's use the signs from one of the options and evaluate the left side of the equation to see if it equals 4.

Let's consider the sequence of signs: \(\minus, +, \times, \div, \times, \times, =\). Substituting these signs into the equation gives:

\({[\{(14 - 6) + (2 \times 3)\} \div (1 \times 7)] \times 2 = 4}\)

Step-by-Step Evaluation Using BODMAS/PEMDAS

We will evaluate the left side of this equation following the standard order of operations (Brackets/Parentheses, Orders/Exponents, Division & Multiplication, Addition & Subtraction):

1. Evaluate the expressions inside the innermost parentheses first:
   • Calculate \((14 - 6)\): \(14 - 6 = 8\)
   • Calculate \((2 \times 3)\): \(2 \times 3 = 6\)
   • Calculate \((1 \times 7)\): \(1 \times 7 = 7\)
2. Substitute these calculated values back into the equation:

   \({[\{(8) + (6)\} \div (7)] \times 2 = 4}\)

3. Next, evaluate the expression within the curly braces \(\{\}\):
   • Calculate \(\{(8) + (6)\}\): \(8 + 6 = 14\)
4. Substitute this value back:

   \({[\{14\} \div (7)] \times 2 = 4}\)

5. Now, evaluate the expression within the square brackets \(\lbrack\rbrack\):
   • Calculate \([\text{14} \div \text{7}]\): \(14 \div 7 = 2\)
6. Substitute this result back into the equation:

   \({2 \times 2 = 4}\)

7. Finally, perform the multiplication on the left side:
   • Calculate \({2 \times 2}\): \(2 \times 2 = 4\)

After performing all operations on the left side, we get the value 4. The equation becomes:

\({4 = 4}\)

Since the left side of the equation is equal to the right side, the equation is balanced with the sequence of signs \(\minus, +, \times, \div, \times, \times, =\).

The correspondence between the '@' positions and the signs used is:

This sequence of signs successfully balances the given equation, making the statement true.

Revision Table: Key Concepts for Equation Balancing

Additional Information: Importance of Brackets in Expressions

Brackets, parentheses, and braces are used in mathematical expressions to group terms and dictate the order in which operations should be performed, overriding the default BODMAS/PEMDAS order if necessary. Operations within the innermost brackets are always performed first.

• Parentheses (): Used for the innermost grouping.
• Curly Braces {}: Used for the next level of grouping outside parentheses.
• Square Brackets []: Used for the outermost level of grouping in complex expressions.

In the given problem, the structure \({[\{(...)\}] \dots}\) clearly shows the nesting levels, guiding the evaluation process from the inside out. Ignoring or misinterpreting the brackets would lead to incorrect results when evaluating the expression and attempting to balance the equation.