Problem 64 - Algebra for College Students

Question

How would you help someone solve the equation $$ \frac{3}{x}-\frac{4}{x}=\frac{-1}{x} \text { ? } $$

Step-by-Step Solution

Answer

The equation is an identity; it holds for all \( x \neq 0 \).

Step 1: Understand the Equation

The equation provided is \( \frac{3}{x} - \frac{4}{x} = \frac{-1}{x} \). This equation involves fractions with a common denominator \( x \).

Step 2: Combine the Fractions

Since all terms in the equation have the same denominator, \( x \), you can subtract the numerators directly. Thus, \( \frac{3}{x} - \frac{4}{x} = \frac{3-4}{x} \).

Step 3: Simplify the Equation

Simplify the numerator by performing the subtraction: \( 3 - 4 = -1 \). This simplifies the left side of the equation to \( \frac{-1}{x} \).

Step 4: Solve the Simplified Equation

Now the equation looks like \( \frac{-1}{x} = \frac{-1}{x} \), which is true for all \( x eq 0 \) because \( x \) cannot be zero in a denominator.

Key Concepts

Fractions with Common Denominators

In algebra, working with fractions often involves ensuring all terms share the same denominator. This allows you to perform operations like addition and subtraction easily. For example, in the equation \( \frac{3}{x} - \frac{4}{x} = \frac{-1}{x} \), each term's denominator is \( x \).

Having a common denominator simplifies tasks because it lets you focus directly on manipulating the numerators. Imagine if the denominators were different; you'd need to find a common denominator first. Think of it like having two slices of pizza: it's straightforward to see that \( \frac{3}{8} - \frac{1}{8} \) just involves subtracting the pizza toppings (or the numerators) while the crust (or the denominator) stays the same.

With common denominators, simply work with the numerators. This is because the denominator acts like a common factor across all fractions involved, streamlining the calculation process.

Simplifying Expressions

Simplifying expressions is a key step in solving equations, especially those involving fractions. In our example, once we recognized the common denominator, the next task was to simplify the expression \( \frac{3}{x} - \frac{4}{x} \).

When simplifying, you're usually looking to combine terms or break them down to their most basic form. This can involve arithmetic operations, like our case where we subtracted the numerators: \( 3 - 4 = -1 \). The expression then simplified to \( \frac{-1}{x} \).

By simplifying expressions, not only do we make equations easier to handle but we also put ourselves in a strong position to recognize equalities, identities, or contradictions in equations. Simplifying expressions reduces complexity and can often reveal the 'hidden' solution more clearly.

Algebraic Manipulation

Algebraic manipulation is about using basic algebraic principles to rearrange and simplify equations. In our solved equation, every step involved such manipulations to achieve clarity.

First, by recognizing the common denominator \( x \), we immediately were able to focus on the numerators' interaction. We applied subtraction directly, leading to a simplified version of our equation. These manipulations were seamless because they employed straightforward strategies.

• Identify common denominators.
• Simplify wherever possible.
• Consistently evaluate and rewrite equations to reduce complexity.

The conclusion showed that both sides of our manipulated equation were equivalent: \( \frac{-1}{x} = \frac{-1}{x} \). This illustrates a basic principle that valid manipulations preserve equality but simplify the equation's form. Algebraic manipulation does not just solve equations; it equips you to tackle a wide array of mathematical challenges effectively.