Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve for \(x: \frac{x}{2}+7=13-\frac{x}{4}\) (Section \(2.3,\) Example 4 )

Step-by-Step Solution

Answer

The solution for \(x\) is 8.

Step 1: Clear the fraction

Multiply each term by 4 to clear the fractions from the equation. This gives: \(2x+28=52-x\).

Step 2: Consolidate x terms on one side

Add \(x\) to both sides of the equation to get only \(x\) terms on one side: \(2x+x+28=52\). This simplifies to: \(3x+28=52\).

Step 3: Isolate x

Finally, subtract 28 from both sides of the equation to solve for \(x\). The equation simplifies to \(3x=52-28\), which gives \(3x=24\). Dividing both sides by 3 results in \(x=8\).

Key Concepts

Algebraic Equations

Algebraic equations are the cornerstone of algebra and are essentially mathematical statements that assert the equality of two expressions. They comprise variables and constants. For example, the equation provided, \( \frac{x}{2} + 7 = 13 - \frac{x}{4} \), is a linear equation where the variable \(x\) represents an unknown value we aim to determine.

When tackling an algebraic equation, it's crucial to apply operations consistently to both sides of the equation to maintain its balance. Imagine a scale in equilibrium; whatever you do to one side must be replicated on the other to preserve this equilibrium. The objective is to perform a series of legal moves to simplify the equation progressively until the variable is easily identifiable and its value can be ascertained.

Isolate Variable

Isolation of the variable is a method employed in solving algebraic equations. The ultimate goal is to get the variable on one side of the equation, standing alone, and all the numbers on the other side. This step is crucial because it transforms an equation from a puzzle into a clear statement of the variable's value.

In the example \(3x + 28 = 52\), we notice that \(x\) is not alone; it's being increased by 28. To segregate \(x\), we must get rid of this 28, and we achieve that by subtracting 28 from both sides of the equation, leading to \(3x = 24\). Notice how each action we take moves us closer to having the variable isolated and easy to identify. Subsequently, we divide by 3 to fully isolate \(x\), resulting in the simple and elegant solution \(x = 8\).

Clearing Fractions

Clearing fractions from an equation makes it more manageable and cleaner to handle. This tactic requires every term in the equation to be multiplied by the same number, usually the least common multiple (LCM) of the denominators, to eliminate these troublesome fractions.

For instance, our starting equation \( \frac{x}{2} + 7 = 13 - \frac{x}{4} \) has denominators of 2 and 4. The LCM of 2 and 4 is 4. Multiplying every term by 4, the LCM, we are able to clear the fractions, converting the equation to \(2x + 28 = 52 - x\). This step doesn't change the equation's solutions; it only simplifies its appearance by ridding it of fractions, making subsequent steps, like isolating the variable, less error-prone and clearer.