- Moira correctly simplified the constant terms, arriving at \(12x = 44 - 10x\).
- The error occurs when moving the variable term: she subtracted \(10x\) instead of adding \(10x\) to both sides.
- Correctly adding \(10x\) leads to the equation \(22x = 44\).
- Solving this correctly results in \(\boxed{x=2}\).
Explanation
- Analyzing the Error
The problem asks us to identify the error in the following algebraic solution:
\[ \begin{aligned} 12x + 10 & = 54 - 10x \ 12x + 10 - 10 & = 54 - 10x - 10 \ 12x & = 44 - 10x \ 12x - 10x & = 44 - 10x + 10x \ 2x & = 44 \ x & = 22 \end{aligned} \]
Let's trace the steps. The first three lines are correct: subtracting 10 from both sides results in \(12x = 44 - 10x\). However, in the fourth line, to isolate terms with \(x\) on one side, we need to add \(10x\) to both sides of the equation. Moira instead subtracts \(10x\) from the left side and adds \(10x\) to the right side, which is inconsistent. She essentially performed \(12x - 10x\) on the left instead of \(12x + 10x\).
- Correcting the Algebraic Operation
To move the \(-10x\) term from the right side of the equation \(12x = 44 - 10x\) to the left, we must perform the inverse operation, which is addition. Adding \(10x\) to both sides of the equation yields:
\[12x + 10x = 44 - 10x + 10x\]
\[22x = 44\]
- Solving and Verifying the Correct Solution
Now that we have the simplified equation \(22x = 44\), we solve for \(x\) by dividing both sides by 22:
\[x = \frac{44}{22} = 2\]
Therefore, the correct value for \(x\) is 2. We can verify this by plugging \(x=2\) into the original equation:
\[12(2) + 10 = 24 + 10 = 34\]
\[54 - 10(2) = 54 - 20 = 34\]
Since both sides result in 34, the solution \(x=2\) is correct.
Examples
Imagine you are managing inventory for two stores. Store A starts with 12 boxes and adds 10 extra items, while Store B starts with 54 items and loses 10 items for every shipment they send out. If you need to find the number of shipments (\(x\)) where both stores have the same total amount of items, you would model the situation with an equation like the one above. Solving it accurately ensures that your distribution logic is correct and both stores remain synchronized in their stock levels.