Problem 9 - Algebra 2

Question

Multiply. State any restrictions on the variables. $$ \frac{8 y-4}{10 y-5} \cdot \frac{5 y-15}{3 y-9} $$

Step-by-Step Solution

Answer

The result of the multiplication is \(\frac{4}{3}\) and the restrictions on \(y\) are that \(y\) cannot be \(0.5\) or \(3\).

Step 1: Identification of Restrictions

Set the denominators of the fractions equal to zero and solve for \(y\). This gives the restrictions on \(y\). In this task, the denominators are \(10y-5=0\) and \(3y-9=0\). Solving these equations gives \(y = 0.5\) and \(y = 3\). So, \(y\) can be any real number except \(0.5\) and \(3\).

Step 2: Simplify the Fractions

The fractions can be simplified further before multiplying. \(\frac{8 y-4}{10 y-5} = \frac{4(2y-1)}{5(2y-1)}\) simplifies to \(\frac{4}{5}\) and \(\frac{5 y-15}{3 y-9} = \frac{5(y-3)}{3(y-3)}\) simplifies to \(\frac{5}{3}\).

Step 3: Multiply the Simplified Fractions

Now multiply the simplified fractions together. \(\frac{4}{5} * \frac{5}{3} = \frac{20}{15}\), which simplifies to \(\frac{4}{3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential step in multiplying fractions efficiently. It involves reducing fractions to their simplest form, meaning the numerator and the denominator have no common factors other than 1. This makes calculations easier and less prone to error.

To simplify a fraction like \(\frac{8y-4}{10y-5}\), look for common factors in the numerator and the denominator. In this case, both parts of the numerator \(8y - 4\) and denominator \(10y - 5\) can be factored as \(4(2y-1)\) and \(5(2y-1)\), respectively. Since \(2y-1\) is a common factor, it can be canceled out, leaving you with \(\frac{4}{5}\).

Applying the same process to another fraction, say \(\frac{5y-15}{3y-9}\), you can factor it to \(\frac{5(y-3)}{3(y-3)}\). The common factor \((y-3)\) cancels out, simplifying the fraction to \(\frac{5}{3}\).

Both fractions are now in their simplest forms, making further operations straightforward.

Identifying Restrictions

Identifying restrictions is a crucial step when working with algebraic fractions to prevent any undefined expression. A fraction is undefined whenever its denominator equals zero, so it's important to determine which values of the variable might cause this.

For this reason, examine the denominators of the given fractions: \(10y-5\) and \(3y-9\). Set them equal to zero and solve for \(y\) to find these restrictions:

• For \(10y-5=0\): solving gives \(y=0.5\).
• For \(3y-9=0\): solving gives \(y=3\).

Thus, the variable \(y\) cannot take values \(0.5\) or \(3\), as these would make the denominators zero, leading to undefined mathematical expressions.

Algebraic Expressions

Algebraic expressions are used everywhere in mathematics, and they play a crucial role in multiplying fractions. An algebraic expression contains variables, numbers, and operations like addition and multiplication. Understanding how to manipulate these expressions is key to working with algebraic fractions.

In the exercise, the expressions in the numerators and denominators are: \(8y-4\), \(10y-5\), \(5y-15\), and \(3y-9\). To simplify these expressions, look for common factors and observe how they can be rewritten in a way that reveals these factors.

Through factoring, complex expressions such as \(8y-4\) and \(10y-5\) become simpler, as they are transformed into \(4(2y-1)\) and \(5(2y-1)\), respectively. Factoring aids in identifying shared components that can be canceled, essential for simplification and ease of operations.

Hence, understanding algebraic expressions thoroughly enhances your ability to simplify, manipulate, and solve problems involving fractions.