Problem 18 - Elementary Algebra

Question

Add or Subtract the following rational expressions. $$ \frac{5 x}{4-x}+\frac{7 x}{x-4} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions and simplify the result: $$\frac{5x}{4-x} + \frac{7x}{x-4}$$ Answer: $$\frac{-2x}{4-x}$$

Step 1: Find the common denominator

To find the common denominator, we can rewrite the two given expressions. Rewrite the second expression by factoring out a -1. This will give us identical denominators for both expressions: $$ \frac{5x}{4-x} + \frac{7x}{-(4-x)} $$ The common denominator is \((4-x)\).

Step 2: Rewrite the rational expressions with the same denominator

Rewrite the rational expressions so that both expressions have the common denominator \((4-x)\): $$ \frac{5x}{4-x} - \frac{7x}{4-x} $$

Step 3: Add the rational expressions

Add the two rational expressions by adding the numerators and keeping the common denominator: $$ \frac{5x - 7x}{4-x} $$

Step 4: Simplify the expression

Simplify the expression by combining like terms in the numerator: $$ \frac{-2x}{4-x} $$ The final answer is: $$ \frac{-2x}{4-x} $$

Key Concepts

Understanding the Common Denominator

When working with rational expressions, one key step is finding a common denominator. This is similar to finding a common denominator in regular fractions. The common denominator allows us to combine expressions easily. For the given rational expressions \(\frac{5x}{4-x}\) and \(\frac{7x}{x-4}\), we need to rewrite them to have the same denominator. However, the denominators \(4-x\) and \(x-4\) are almost the same, except one is the negative of the other. By factoring out \(-1\) from \(x-4\), we can rewrite it as \(-(4-x)\). Now, both expressions have a common denominator \(4-x\), enabling us to complete the arithmetic operation. This step aligns the expressions so they can be easily added or subtracted.

Factoring in Rational Expressions

Factoring is a key tool in simplifying and solving mathematical expressions. In rational expressions, it helps to adjust denominators and numerators to find commonality or simplify expressions. Often, we factor to reveal or exploit patterns that may simplify our problem. For example, with the denominator \(x-4\), when rewritten as \(-(4-x)\), we use the technique of factoring out a negative to make it similar to the other denominator, \(4-x\). Factoring can reveal hidden common denominators, making it easier to work with rational expressions and allowing us to rewrite them in a more manageable form.

Combining Like Terms

Once expressions are rewritten with a common denominator, the next step is to combine terms that are similar. These are known as 'like terms'. In the numerator of the expression \(\frac{5x}{4-x} - \frac{7x}{4-x}\), both terms share the variable \(x\). To combine like terms, simply operate on the coefficients (the numbers in front of \(x\)). Thus, you subtract \(7x\) from \(5x\), resulting in \(-2x\). Combining like terms simplifies the expression and reduces the expression to its simplest form, making further calculations easier.

Simplifying Algebraic Expressions

The last step in handling rational expressions is simplifying algebraic expressions. Simplification involves reducing expressions to their most concise and straightforward form. After combining like terms, the expression becomes \(\frac{-2x}{4-x}\). This expression is already quite simple. Simplifying includes making sure there are no further common factors between the numerator and denominator that can be divided out. For this specific expression, there are no common factors to simplify further. Therefore, \(\frac{-2x}{4-x}\) is the simplest form of the expression. Simplification is crucial as it makes equations easier to understand and solve in algebra.