Problem 91 - Elementary Algebra

Question

Explain to a classmate why this is incorrect: \(1 \times 2+2 \times 2=32 \times 2 .\)

Step-by-Step Solution

Answer

The error is in equality: 6 does not equal 64.

Step 1: Evaluate Each Side Separately

To understand the error, we first evaluate each side of the equation separately. Start by calculating the left-hand side: \(1 \times 2 + 2 \times 2\). These follow: \((1 \times 2) = 2\) and \((2 \times 2) = 4\). So, the left-hand side is \(2 + 4 = 6\).

Step 2: Evaluate the Right-Hand Side

Now evaluate the right-hand side: \(32 \times 2\). Multiplication gives us \(32 \times 2 = 64\).

Step 3: Compare Both Sides

After evaluating both sides, compare them to see if they are equal. From Step 1, the left-hand side is 6, and from Step 2, the right-hand side is 64. These values are not equal.

Step 4: Identify and Explain the Error

Explain the mistake: the initial equation stated that \(1 \times 2 + 2 \times 2 = 32 \times 2\), but calculations showed this is incorrect because 6 is not equal to 64. This likely arose from a calculation or transcription error.

Key Concepts

Equations

An equation is a mathematical statement that shows the equality of two expressions. It is vital to handle each component correctly to solve it accurately.
In the given exercise, the equation is:

• Left-hand side: \( 1 \times 2 + 2 \times 2 \)
• Right-hand side: \( 32 \times 2 \)

When evaluating an equation, each side must be considered separately. Then, the values obtained from both sides should be compared to check if they are truly equivalent. Understanding how to analyze each side separately ensures you won't mix up operations or values inadvertently.
Observing the result of both sides (6 and 64) clarifies that they are not equal, hence proving the equation given is incorrect.

Multiplication

Multiplication is one of the fundamental operations in algebra. It involves repeated addition and is essential for evaluating expressions accurately.
Let's revisit the original problem:

• The left side requires multiplying twice: \( 1 \times 2 = 2 \) and \( 2 \times 2 = 4 \). So, \( 2 + 4 = 6 \).
• The right side simplifies to \( 32 \times 2 = 64 \).

Proper handling of multiplication ensures the accuracy of results. Notice how each pair of numbers was multiplied independently on the left side, and the resulting products added together. In algebra, always follow the order of operations to avoid errors. Missteps commonly arise when the multiplication is not computed correctly or the process is rushed.

Error analysis

Error analysis is a valuable skill that involves reviewing and understanding where a mistake occurred. It is fundamental in learning from incorrect solutions.
In the exercise, the error was identified by

• Evaluating each side separately.
• Comparing the calculated results.

The given input erroneously stated both sides to be equal. However, analytical checking revealed the mismatch in numerical values. Such errors could result from

• Misreading operation symbols or numbers.
• Performing multiplication or addition inaccurately.

Correcting these issues involves double-checking each step to ensure alignment between operations and expected results. By practicing this vigilant checking, accuracy in solving equations improves significantly.