Problem 70 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations $$\frac{x^{2}-x z+x y-y z}{x-y}+\frac{x-z}{y-x}$$

Step-by-Step Solution

Answer

The simplified expression of the given operation is \(\frac{(x^{2}+x(y-z)+(z(1-y))}{(x-y)}\).

Step 1: Find the common denominator

Given are two fractions, \(\frac{x^{2}-x z+x y-y z}{x-y}\) and \(\frac{x-z}{y-x}\). To add them, we first need a common denominator. Notice that the denominators \(x-y\) and \(y-x\) are negatives of each other. Thus, we can multiply the second expression by \(-1/-1\) to make the denominators same, which changes second expression to \(\frac{-(x-z)}{-(y-x)}\) or \(\frac{(z-x)}{(x-y)}\).

Step 2: Add the fractions

Now that both fractions have the same denominator, they can be added together. \(\frac{(x^{2}-x z+x y-y z) + (z-x)}{(x-y)}\). This simplifies to \(\frac{(x^{2}+x y+z-x-x z-y z)}{(x-y)}\).

Step 3: Simplify the fraction

On further simplifying the expression we get, \(\frac{(x^{2}+x y-x z+z-y z)}{(x-y)} = \frac{(x^{2}+x (y-z)+(z-y z))}{(x-y)} = \frac{(x^{2}+x(y-z)+(z(1-y))}{(x-y)}\).

Key Concepts

Fractions

Fractions involve two parts: a numerator, which is the top part, and a denominator, the bottom part. The fraction itself represents a division of the numerator by the denominator.
When working with algebraic fractions, these numerators and denominators can include variables, such as in the examples \(\frac{x^{2}-x z+x y-y z}{x-y}\) and \(\frac{x-z}{y-x}\).
The common goal with fractions in algebra is to make calculations easier or to combine them. In our exercise, for instance, we needed to add two fractions together.
Understanding how to handle these variable expressions and manipulate them into a common format is crucial for simplifying and solving problems.

Common Denominator

One of the crucial steps in combining or adding fractions is finding a common denominator. This is important because it allows us to operate on each fraction as if they were part of the same division scheme.
In the given exercise, we noticed denominators \(x-y\) and \(y-x\) are negatives of each other. By multiplying one fraction by \(-1\), we can alter its form without changing its value, transforming \(\frac{x-z}{y-x}\) into \(\frac{z-x}{x-y}\) so that both expressions have the denominator \(x-y\).
Finding common denominators is a technique that helps bring different algebraic fractions into a form where they can be easily added or subtracted.

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms and reducing the expression to its simplest form. The aim is to make the expression easier to understand or solve.
Once fractions have a common denominator, like in our example, we can add the numerators directly: \((x^2 - xz + xy - yz) + (z-x)\).
Next, we combine like terms. Those terms with the same base variable or coefficient are grouped together and simplified, resulting in \(x^{2}+x(y-z)+(z(1-y))\) over the common denominator \(x-y\).
Doing so reduces complexity and reveals the core structure of the equation, thereby assisting with any further mathematical operations or insights needed for solving the problem at hand.