Problem 25 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{x^{2}+x}{5} \cdot \frac{25}{x y+y}$$

Step-by-Step Solution

Answer

\( \frac{5x}{y} \).

Step 1: Simplify Each Fraction

First, simplify the numerators and denominators of each fraction where possible. The first fraction \( \frac{x^2 + x}{5} \) can be simplified by factoring out an \( x \) in the numerator: \( x(x+1) \). The second fraction \( \frac{25}{xy + y} \) simplifies by factoring \( y \) out of the denominator: \( y(x+1) \).

Step 2: Write the Product of Fractions

Now, express the multiplication of the two simplified fractions: \[ \frac{x(x+1)}{5} \cdot \frac{25}{y(x+1)} \].

Step 3: Multiply the Numerators

Multiply the numerators of the two fractions: \( x(x+1) \times 25 = 25x(x+1) \).

Step 4: Multiply the Denominators

Multiply the denominators of the two fractions: \( 5 \times y(x+1) = 5y(x+1) \).

Step 5: Write the Combined Fraction

Combine both results into a single fraction: \[ \frac{25x(x+1)}{5y(x+1)} \].

Step 6: Simplify the Combined Fraction

Cancel out the common factor \( x+1 \) from the numerator and the denominator: \[ \frac{25x}{5y} \].

Step 7: Simplify Further if Possible

Now, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5: \( \frac{25x}{5y} = \frac{5x}{y} \).

Key Concepts

Factoring

Factoring is an essential skill in algebra that involves expressing a mathematical expression as a product of its factors. In the context of this exercise, factoring allows us to simplify complex expressions, making them more manageable and easier to work with.

• Consider the expression \(x^2 + x\). By factoring, we look for a common factor in this expression. Here, both terms contain an \(x\), so we factor it out, yielding \(x(x+1)\).
• Similarly, for the expression \(xy + y\) in the denominator, the common factor is \(y\), resulting in \(y(x+1)\).

Factoring not only simplifies expressions but also reveals common factors that can potentially be canceled out in fractions. This technique not only streamlines calculations but also enhances clarity in mathematical problem-solving.

Fractions

Fractions represent parts of a whole and consist of a numerator over a denominator. In algebra, understanding how to manipulate fractions is crucial, particularly with expressions that involve variables.
Working with algebraic fractions involves several steps:

• First, simplify the fractions by factoring where possible, as done when transforming \(\frac{x^2 + x}{5}\) into \(\frac{x(x+1)}{5}\).
• Next, multiply two or more fractions by multiplying their numerators together and their denominators together, effectively creating a new, single fraction. For example, combining \(\frac{x(x+1)}{5}\) and \(\frac{25}{y(x+1)}\) results in \(\frac{25x(x+1)}{5y(x+1)}\).
• Finally, simplify the resulting fraction by canceling any common factors shared between the numerator and the denominator.

Practicing these steps with algebraic fractions builds a solid foundation for handling more complicated algebraic equations and operations.

Simplification

Simplification in algebra is the process of making an expression as straightforward as possible. This typically involves reducing the expression to have no unnecessary terms or factors, which can lead to clearer solutions and results.
The simplification process generally includes:

• Identifying and canceling common factors in both the numerator and the denominator of a fraction, such as the term \(x+1\) in our example.
• Further reducing the expression by dividing every term by their greatest common divisor. This can be seen when \(\frac{25x}{5y}\) simplifies to \(\frac{5x}{y}\) since both numbers have a common divisor of 5.

Simplification helps to produce the simplest, clearest version of an expression for ease of interpretation and further mathematical operations. By simplifying expressions, students can focus on understanding the core mathematical concepts without being distracted by complex, unwieldy equations.