- Moira correctly subtracts 10 from both sides but makes an error by subtracting \(10x\) from the left side instead of adding \(10x\).
- The correct procedure is to add \(10x\) to both sides, which simplifies the equation to \(22x + 10 = 54\).
- Subtracting 10 from both sides yields \(22x = 44\).
- Dividing by 22 results in the correct solution of \(\boxed{x = 2}\).
Explanation
- Analyzing Moira's Error
The equation to solve is \(12x + 10 = 54 - 10x\). Let's examine the steps taken by Moira to solve this:
- \(12x + 10 = 54 - 10x\)
- \(12x + 10 - 10 = 54 - 10x - 10\) (Subtract 10 from both sides) -> \(12x = 44 - 10x\) (This step is correct).
- \(12x - 10x = 44 - 10x + 10x\) (She intended to add \(10x\) to both sides, but on the left side, she performed subtraction \(12x - 10x\) instead of addition \(12x + 10x\)).
- \(2x = 44\) (Result of the previous incorrect step).
- \(x = 22\) (Calculated as \(44 / 2\)).
The error occurred in the transition from \(12x = 44 - 10x\) to the next step. To isolate the \(x\) terms on one side, one must add \(10x\) to both sides. Moira wrote \(12x - 10x\) on the left side, effectively subtracting \(10x\) instead of adding it. This led to an incorrect simplification of \(2x\) instead of \(22x\).
- Correcting the Algebra
To solve the equation correctly, we must gather all terms involving \(x\) on one side and constant terms on the other:
- Start with the equation: \(12x + 10 = 54 - 10x\)
- Add \(10x\) to both sides to move all \(x\) terms to the left:
\(12x + 10x + 10 = 54 - 10x + 10x\)
\(22x + 10 = 54\)
- Subtract 10 from both sides to isolate the \(22x\) term:
\(22x + 10 - 10 = 54 - 10\)
\(22x = 44\)
- Divide both sides by 22 to solve for \(x\):
\(x = \frac{44}{22} = 2\)
Verifying the result: Substituting \(x = 2\) into the original equation:
Left Side: \(12(2) + 10 = 24 + 10 = 34\)
Right Side: \(54 - 10(2) = 54 - 20 = 34\)
Since the left and right sides match, the correct solution is \(x = 2\).
Examples
Solving linear equations is essential for daily tasks like budgeting. For example, if you are comparing two payment plans, one with a base fee plus an hourly rate and another with a higher base fee but a lower hourly rate, you can set them equal to each other to find the "break-even" point. Just as in our problem, correctly isolating your variables ensures you find the exact time or cost where both options are equal, helping you make informed financial decisions.