Equation Correction by Interchanging Signs and Numbers

The problem asks us to identify which pair of sign changes and number changes correctly solves the given mathematical equation. We need to find the specific interchanges that make the equation true.

The initial equation provided is:

$10 + 4 - 5 \times 2 = 19$

Our task is to test the given options to see which one satisfies this equation after performing the specified interchanges.

Analyzing Option 2: Signs '+' and '$\times$', Numbers '4' and '2'

Option 2 suggests the following interchanges:

• Signs to interchange: The plus sign ($+$) and the multiplication sign ($\times$).
• Numbers to interchange: The number 4 and the number 2.

Let's apply these changes step-by-step to the original equation:

Original equation: $10 + 4 - 5 \times 2 = 19$

Step 1: Interchange the signs '+' and '$\times$'.

• The '+' sign located between 10 and 4 is replaced with '$\times$'.
• The '$\times$' sign located between 5 and 2 is replaced with '+'.
• The '-' sign remains unchanged.
• After this sign interchange, the equation becomes: $10 \times 4 - 5 + 2 = 19$

Step 2: Interchange the numbers '4' and '2'.

• The number '4' (which is now after the first operator) is replaced with '2'.
• The number '2' (which is now at the end of the expression) is replaced with '4'.
• After this number interchange, the equation becomes: $10 \times 2 - 5 + 4 = 19$

Verification of the Corrected Equation

To confirm if this corrected equation is true, we evaluate the left-hand side (LHS) using the standard order of operations (PEMDAS/BODMAS):

The corrected equation is $10 \times 2 - 5 + 4 = 19$.

1. Perform multiplication first: $10 \times 2 = 20$.
2. The expression simplifies to: $20 - 5 + 4$.
3. Perform subtraction next (from left to right): $20 - 5 = 15$.
4. The expression simplifies to: $15 + 4$.
5. Perform addition last: $15 + 4 = 19$.

The calculated value of the LHS is $19$. This matches the right-hand side (RHS) of the original equation ($19$). Therefore, interchanging the signs '+' and '$\times$', along with the numbers '4' and '2', correctly solves the equation.