Solving Equations by Swapping Numbers

The question requires us to find a pair of numbers from the given options that, when swapped in the original equation, makes the equation true. The original equation is:

$\qquad 126 \div 23 - 138 \div (5 \times 4 - 6) + 4 = 1$

We will examine each option by swapping the numbers and evaluating the left side of the equation to see if it equals 1.

Testing Option 1: Swapping 4 and 6

If we interchange the numbers 4 and 6 in the equation, the new equation becomes:

$\qquad 126 \div 23 - 138 \div (5 \times 6 - 4) + 6$

Evaluating the left side following the order of operations (BODMAS/PEMDAS):

$\qquad = 126 \div 23 - 138 \div (30 - 4) + 6$

$\qquad = 126 \div 23 - 138 \div 26 + 6$

Performing the divisions (approximately):

$\qquad \approx 5.478 - 5.308 + 6$

Performing subtraction and addition:

$\qquad \approx 0.17 + 6$

$\qquad \approx 6.17$

Since $6.17 \ne 1$, swapping 4 and 6 does not make the equation correct.

Testing Option 2: Swapping 138 and 126

Let's interchange the numbers 138 and 126 in the equation. The new equation is:

$\qquad 138 \div 23 - 126 \div (5 \times 4 - 6) + 4$

Now, let's evaluate the left side step-by-step using the order of operations:

First, calculate the expression inside the parentheses:

$\qquad (5 \times 4 - 6) = (20 - 6) = 14$

Substitute this value back into the equation:

$\qquad 138 \div 23 - 126 \div 14 + 4$

Next, perform the divisions from left to right:

$\qquad 138 \div 23 = 6$

$\qquad 126 \div 14 = 9$

Substitute these results:

$\qquad 6 - 9 + 4$

Finally, perform the subtraction and addition from left to right:

$\qquad (6 - 9) + 4 = -3 + 4 = 1$

The left side evaluates exactly to 1, which is the value on the right side of the original equation. Thus, swapping 138 and 126 makes the equation correct.

Testing Option 3: Swapping 4 and 1

Interpreting this as swapping the number 4 on the left side with the target result 1 on the right side, the equation we check if the original left side with 4 replaced by 1 equals the original right side's 1 replaced by 4. The left side becomes:

$\qquad 126 \div 23 - 138 \div (5 \times 4 - 6) + 1$

The equation we are checking becomes:

$\qquad 126 \div 23 - 138 \div (5 \times 4 - 6) + 1 = 4$

We already know $(5 \times 4 - 6) = 14$. So the left side evaluation is:

$\qquad = 126 \div 23 - 138 \div 14 + 1$

$\qquad \approx 5.478 - 9.857 + 1$

$\qquad \approx -4.379 + 1$

$\qquad \approx -3.379$

Since $-3.379 \ne 4$, swapping 4 and 1 does not make the equation correct under this interpretation.

Testing Option 4: Swapping 23 and 138

If we interchange the numbers 23 and 138 in the equation, the new equation becomes:

$\qquad 126 \div 138 - 23 \div (5 \times 4 - 6) + 4$

Evaluating the left side:

$\qquad = 126 \div 138 - 23 \div (20 - 6) + 4$

$\qquad = 126 \div 138 - 23 \div 14 + 4$

Performing the divisions (approximately):

$\qquad \approx 0.913 - 1.643 + 4$

Performing subtraction and addition:

$\qquad \approx -0.73 + 4$

$\qquad \approx 3.27$

Since $3.27 \ne 1$, swapping 23 and 138 does not make the equation correct.

Summary of Number Swaps and Results

The table below summarizes the result of each interchange option:

Additional Information: Order of Operations

The correct evaluation of mathematical expressions depends on following the standard order of operations. This ensures a consistent result for any given expression. A common acronym used to remember the order is BODMAS or PEMDAS.

• B / P: Brackets or Parentheses - Operations inside grouping symbols are performed first.
• O / E: Orders or Exponents - Powers and roots are calculated next.
• DM: Division and Multiplication - These are done from left to right.
• AS: Addition and Subtraction - These are done from left to right after division and multiplication.

In this problem, applying the order of operations was essential to correctly evaluate the equation after each proposed number swap.