Problem 79 - Elementary Algebra

Question

What is wrong with the following procedure? Explain how it should be done. $$ \begin{aligned} \frac{x}{2}+4 &=\frac{x}{6} \\ 6\left(\frac{x}{2}+4\right) &=2(x) \\ 3 x+24 &=2 x \\ x &=-24 \end{aligned} $$

Step-by-Step Solution

Answer

The error was in the multiplication step; the correct solution is \( x = -12 \).

Step 1: Identify Potential Issues

Let's examine each step of the procedure. We start with the equation \( \frac{x}{2} + 4 = \frac{x}{6} \). Observe how we try to remove fractions by multiplying the entire equation with the least common multiple of the denominators (2 and 6), which is 6.

Step 2: Correct the Multiplication Step

Multiplying every term by 6, we get: \( 6 \cdot \frac{x}{2} + 6 \times 4 = 6 \cdot \frac{x}{6} \). This resolves to \( 3x + 24 = x \) instead of \( 2x = 3x + 24 \).

Step 3: Solve the Corrected Equation

With the corrected equation \( 3x + 24 = x \), subtract \( x \) from both sides to get \( 2x + 24 = 0 \).

Step 4: Isolate the Variable

Subtract 24 from both sides, yielding \( 2x = -24 \).

Step 5: Divide to Solve for x

Divide both sides by 2 to solve for \( x \): \( x = -12 \).

Key Concepts

Understanding Equation Correction

When solving an equation, we often need to address mistakes to find the correct solution. Equation correction is the process of identifying errors in a mathematical procedure and making the necessary amendments.
In our given problem, the key mistake lies in the step where fractions were eliminated. It is essential to properly distribute the multiplying factor to every term in the equation. This helps maintain the equation's balance and ensures that the calculations reflect the equation's original intentions. By multiplying both sides by the least common multiple of the denominators, we effectively "clear" the fractions, simplifying the equation.
In this specific exercise, the equation was initially set as \( \frac{x}{2} + 4 = \frac{x}{6} \). Multiplying each term by 6 correctly results in the simpler equation \( 3x + 24 = x \). Addressing the error early on avoids further complications and ensures accurate solutions later in the process.

Mastering Fraction Multiplication

Fraction multiplication might seem straightforward, but it's crucial to apply it correctly when solving equations. This operation involves multiplying a fraction by a number or another fraction, affecting every term in the equation.
When dealing with equations containing fractions, it is often necessary to eliminate the fractions to make the solving process simpler. As demonstrated in the original problem, multiplying each term by the least common multiple effectively removes the fractions.

• The least common multiple (LCM) of the denominators ensures all fractions are handled with one multiplication.
• It is crucial to apply this multiplication to every term on both sides of the equation to maintain balance.

By learning fraction multiplication strategies, solving equations with complex expressions becomes more approachable.

Variable Isolation Essentials

Variable isolation is a key step in solving equations, as it involves manipulating the equation to have the variable on one side and all other terms on the opposite side. This makes finding the value of a variable straightforward.
In our problem, after correcting the equation to \( 3x + 24 = x \), we isolate x as follows:

• First, subtract \( x \) from both sides, resulting in \( 2x + 24 = 0 \).
• Next, subtract 24 from both sides to yield \( 2x = -24 \).
• Finally, divide both sides by 2 to solve for \( x \), which gives \( x = -12 \).

By using these steps, isolating a variable becomes a more straightforward process. It allows us to efficiently reach the solution while keeping the equation balanced throughout. Mastery of variable isolation techniques is crucial for successfully solving algebra problems.