Problem 44 - Elementary Algebra Within Reach

Question

Error Analysis A student solves the equation \(S=2 l w+2 l h+2 w h\) for \(w\) and his answer is \(w=\frac{S-2 l w-2 l h}{2 h}\). Describe and correct the student's error.

Step-by-Step Solution

Answer

The student's error lies in incorrectly moving terms to solve for \(w\). The correct solution is \(w=\frac{S-2lh}{2l+2h}\).

Step 1: Identify Errors

Check the student's solution. He attempted to solve the equation by moving the two terms involving \(w\) to the left side. This would be correct if we were solving for \(S\), but it is incorrect when solving for \(w\).

Step 2: Correct Approach

Start from the given equation \(S=2lw+2lh+2wh\). As the task is to solve for \(w\), isolate \(w\) associated terms on one side of the equation. Subtract \(2lh\) from both sides, to get \(S-2lh=2lw+2wh\).

Step 3: Solve for \(w\)

Now, factor out \(w\) from the right side yield \(S-2lh=w(2l+2h)\). Finally, divide both sides by \((2l+2h)\) to solve for \(w\), that is \(w=\frac{S-2lh}{2l+2h}\).

Key Concepts

Error Analysis in Algebra

Error analysis is a crucial step in the learning process, particularly when dealing with algebraic equations. This process involves identifying and understanding the mistakes made during problem-solving to prevent similar errors in future attempts. In the presented exercise, the student has made an error by trying to isolate the variable 'w' without correctly manipulating all terms relating to 'w' in the equation. The mistake stems from not comprehensively understanding that to isolate a variable, one must first combine all terms containing that variable on one side of the equation before any other operations are performed.

It's essential to approach error correction constructively, acknowledging that errors are a natural part of learning. To rectify the mistake, the student should first consolidate all terms involving 'w,' and then use inverse operations to isolate 'w' on one side of the equation. A careful step-by-step review of the process helps not only in solving the equation correctly but also reinforces a strategic approach to problem-solving in algebra.

Isolating Variables

Isolating variables is an integral skill in algebra that enables one to find the value of an unknown in an equation. To isolate a variable successfully, you must manipulate the equation in such a way that the variable of interest stands alone on one side of the equal sign, with all other terms on the opposite side. It is akin to finding a 'target' while shifting all 'obstacles' out of the way.

In the given exercise, isolating 'w' necessitates moving all other 'w'-related terms to the same side and collecting like terms. This often involves the use of addition or subtraction to remove unwanted terms from the side with the variable and multiplication or division to remove coefficients attached to the variable itself. It's like untangling a knot — each move should simplify the equation without disrupting the rest. When done correctly, we eventually reach a point where the variable is expressed as a function of other known values or variables, a cornerstone tactic for solving algebraic equations.

Remember the Goal

Keep in mind that the goal is to have the variable of interest completely isolated. This means that all coefficients and unrelated terms should be on the other side of the equation. Every step taken should nudge us closer to this goal.

Factoring Algebraic Expressions

Factoring is a powerful algebraic technique that simplifies expressions and is vital for solving equations efficiently. It involves breaking down an equation into a product of simpler factors, thereby enabling easier manipulation and, ultimately, the isolation of variables.

The student's original error was partially due to a lack of factoring. By not recognizing that 'w' could be factored out from the terms on one side of the equation, they prematurely attempted to move terms across the equal sign. Once all 'w' terms are on the same side, as seen in the corrected approach, factoring 'w' out simplifies the equation to a form where 'w' can be isolated by dividing both sides by the binomial factor (in this case, '2l+2h').

Utilize Factoring as a Tool

Always consider factoring when faced with multiple terms that share a common variable, as it can reduce a seemingly complex equation down to a more manageable one. Factoring lays the groundwork for further steps, including isolating variables, and is a technique that can vastly improve algebraic problem-solving skills. In our context, factoring enables us to neatly separate 'w' from other terms, streamlining the path to the solution.