Solving the Equation for the Value of k

The question asks us to find the value of the variable \(k\) in the given mathematical equation: \(13 - [10 - \{16 + 8 \div (k - 2)\} + 2] = 20\). To solve for \(k\), we need to isolate \(k\) by simplifying the equation step by step, following the order of operations (often remembered by acronyms like BODMAS or PEMDAS).

Step-by-Step Solution to Find k

Let's break down the equation and solve it systematically:

The given equation is:

Step 1: Isolate the main bracket term

First, move the term outside the outermost square bracket to the right side of the equation:

Now, multiply both sides by -1 to remove the negative sign outside the square bracket:

Step 2: Simplify inside the square bracket

Combine the constant terms inside the square bracket (which are now on the left side):

Step 3: Isolate the curly brace term

Move the constant term \(12\) to the right side of the equation:

Multiply both sides by -1 to remove the negative sign outside the curly brace:

Step 4: Isolate the division term

Move the constant term \(16\) to the right side of the equation:

Rewrite the division as a fraction:

Step 5: Solve for k

Multiply both sides of the equation by \((k - 2)\) to remove the denominator:

Add 6 to both sides of the equation:

Finally, divide both sides by 3 to find the value of \(k\):

Thus, the value of \(k\) that satisfies the given equation is \(\frac{14}{3}\).

Verification (Optional but Recommended)

To verify the solution, substitute \(k = \frac{14}{3}\) back into the original equation:

\(13 - [10 - \{16 + 8 \div (\frac{14}{3} - 2)\} + 2]\)

Calculate the term inside the parenthesis:

\(\frac{14}{3} - 2 = \frac{14}{3} - \frac{6}{3} = \frac{14 - 6}{3} = \frac{8}{3}\)

Now substitute this back:

\(13 - [10 - \{16 + 8 \div (\frac{8}{3})\} + 2]\)

Perform the division:

\(8 \div \frac{8}{3} = 8 \times \frac{3}{8} = 3\)

Substitute this back:

\(13 - [10 - \{16 + 3\} + 2]\)

Simplify inside the curly braces:

\(13 - [10 - \{19\} + 2]\)

\(13 - [10 - 19 + 2]\)

Simplify inside the square brackets:

\(10 - 19 + 2 = -9 + 2 = -7\)

Substitute this back:

\(13 - [-7]\)

\(13 + 7 = 20\)

The left side equals 20, which matches the right side of the original equation. Thus, the value \(k = \frac{14}{3}\) is correct.

Revision Table: Equation Solving Steps

Additional Information: Order of Operations (BODMAS/PEMDAS)

Solving equations like this requires strictly following the order of operations. This ensures that we simplify the expression correctly. The acronyms BODMAS and PEMDAS help remember this order:

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

In our problem, we started with the innermost parentheses/brackets `(k-2)`, although we couldn't simplify that directly. We then worked outwards from the curly braces `{}` to the square brackets `[]`, performing operations in the correct order (division before addition/subtraction). Finally, we used inverse operations (subtraction/addition, multiplication/division) to isolate the variable \(k\).