Problem 69 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Express Division as Multiplication

The operation we're asked is a division of fractions. The division of fractions can be transformed into a multiplication operation by multiplying the first fraction by the reciprocal of the second fraction. So, the expression \(\frac{x^{2}+x z+x y+y z}{x-y} \div \frac{x+z}{x+y}\) is equivalent to \(\frac{x^{2}+x z+x y+y z}{x-y} \cdot \frac{x+y}{x+z}\)

Step 2: Simplify the Resulting Expression

At this point, there is no further simplification that can be done. Multiplication and division have the same precedence, so, from left to right, we get the final form of the expression: \(\frac{(x^{2}+x z+x y+y z)(x+y)}{(x-y)(x+z)}\)