To determine the mistake Carlos made while solving the equation, we first need to identify the original equation he was solving. Since we don’t have the equation presented, we can discuss a general approach to how he might have arrived at the solution and common errors to look out for.
Assuming Carlos made a mistake involving an algebraic operation, possible mistakes could include:
- Incorrect arithmetic operations: He might have added or subtracted incorrectly at some point in the process.
- Misapplication of inverse operations: If he was supposed to multiply or divide but confused those operations, that could lead to an incorrect solution.
Checking the solution:
To check if \( x = 6.5 \) is indeed the solution, Carlos should substitute \( 6.5 \) back into the original equation. Here’s a general process for checking:
- Substitute: Replace \( x \) in the original equation with \( 6.5 \).
- Evaluate: Calculate both sides of the equation to see if they are equal.
- Compare: If both sides of the equation are equal after substitution, then \( x = 6.5 \) is the correct solution. If not, then Carlos made a mistake.
Example:
Suppose the equation was \( 2x + 3 = 16 \).
- Carlos solved for \( x \) and found \( x = 6.5 \).
- Checking:
- Substitute \( x \) back: \( 2(6.5) + 3 = 13 + 3 = 16 \).
- The left side equals the right side, so in this case, Carlos would be correct.
If it's incorrect:
If the equation was somewhere else, for instance \( 2x + 3 = 12 \):
- The correct solution would be \( x = 4.5 \).
- If Carlos found \( x = 6.5 \):
- Checking: \( 2(6.5) + 3 = 13 + 3 = 16 \) (not equal to 12).
- Thus, he made a mistake in his solving process (perhaps miscalculating or misapplying operations).
By confirming through substitution, Carlos could identify any mistakes in his solution process and rectify them for future problems.