Problem 56 - Algebra 2

Question

Simplify each expression. \(\frac{3 x}{x-y}+\frac{4 x}{y-x}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-x}{x-y}\).

Step 1: Identify the denominators and factor

First, observe that the denominators are \(x-y\) and \(y-x\). Notice that \(y-x\) can be rewritten as \(-(x-y)\) to factor out a negative sign. This allows us to have a common denominator.

Step 2: Rewrite the expression with a common denominator

Using the identity \(y-x = -(x-y)\), we can rewrite the second fraction's denominator as \(\frac{4x}{-(x-y)}\). Therefore, the expression becomes: \[ \frac{3x}{x-y} - \frac{4x}{x-y} \] Now both fractions have the same denominator \(x-y\).

Step 3: Combine the fractions

Now that both fractions have the same denominator, we can combine them into a single fraction by subtracting the numerators: \[ \frac{3x - 4x}{x-y} = \frac{-x}{x-y} \].

Step 4: Simplify the result

The expression \(\frac{-x}{x-y}\) is already in its simplest form. Therefore, we have successfully simplified the given expression.

Key Concepts

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra where the goal is to remove any unnecessary parts of the expression to make it as concise as possible. For fraction simplification, we often deal with two primary tasks: reducing fractions by dividing the numerator and the denominator by a common factor and combining fractions that have similar terms or denominators.

• Identifying the greatest common factor (GCF) is key. It’s essential to look for numbers or variables that can be factored out from both the numerator and the denominator.
• Once a common factor is found, divide both parts by this factor to simplify the fraction.
• Combining like terms can also simplify a fraction, especially when dealing with algebraic expressions.

It's important to remember that the aim of simplifying is to make the expression easier to understand and work with, without changing its value.

Common Denominators

Finding common denominators is crucial when adding or subtracting fractions because it allows for straightforward computation. The common denominator gives the two fractions the same base, which makes it easier to perform operations on them.

• When fractions have different denominators, they must first be rewritten with a common denominator. This involves finding a shared multiple of the two denominators.
• In algebraic expressions, manipulating denominators often involves factoring them to reveal common elements that can be simplified.
• In some cases, like our current exercise, recognizing a simple sign change or identity can help in quickly establishing a common denominator. As in the example provided, noticing that \(y-x = -(x-y)\) was key to progress in the solution.

Once a common denominator is achieved, the fractions can be combined into a single fraction by adding or subtracting the numerators.

Factoring

Factoring is the process of breaking down an expression into a product of its simpler components, or factors. It is especially useful in simplifying algebraic expressions. Factoring is used to find common factors among terms that can help break down complex expressions into more manageable parts.

• Identifying patterns such as differences of squares, trinomial squares, or recognizing common binomial factors is important for efficient factoring.
• In cases where a factor is negative, it might be necessary to factor out a negative one to standardize denominators, as seen with \(y-x\) being rewritten as \(-(x-y)\).
• Once factored, these expressions can sometimes be further simplified by canceling out common terms, leading to a more elegant or simplified expression.

Factoring is a versatile tool in simplifying fractions and solving algebraic equations.