Problem 50 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(\frac{y}{2 z}\)

Step-by-Step Solution

Answer

The expression evaluates to \(0.3\).

Step 1: Substitute Variables

Replace each variable in the expression with its given value: - Substitute 3 for \(y\). - Substitute 5 for \(z\). The expression \(\frac{y}{2z}\) becomes \( \frac{3}{2 \times 5} \).

Step 2: Calculate Denominator

Multiply the values in the denominator: - Calculate \(2 \times 5 = 10\). Now the expression simplifies to \( \frac{3}{10} \).

Step 3: Evaluate Fraction

As the fraction \( \frac{3}{10} \) is already in its simplest form, the evaluation is complete. Expressed as a decimal, \( \frac{3}{10} = 0.3 \).

Key Concepts

Variable Substitution

Variable substitution is a key concept in algebra that involves replacing variables in an expression with given numerical values. It simplifies solving equations by allowing you to work with numbers instead of abstract symbols. For example, in the expression \( \frac{y}{2z} \), if we are given that \( y = 3 \) and \( z = 5 \), we can substitute these values directly into the expression.

• Start by identifying each variable in your expression.
• Replace every instance of those variables with the specified numbers.

After substitution, \( \frac{y}{2z} \) becomes \( \frac{3}{2 \times 5} \). This handy technique can help solve a wide array of algebraic problems by converting them into straightforward arithmetic tasks.

Fraction Simplification

Simplifying fractions is an essential step to make an expression cleaner and more manageable. It involves reducing the fraction to its simplest form by eliminating any common factors between the numerator and denominator.
In the expression we evaluated, \( \frac{3}{10} \), there are no common factors between 3 and 10 other than 1, meaning the fraction is already simplified.
To determine this:

• Check if there is a number greater than 1 that divides both the numerator and the denominator evenly.
• If no such number exists, the fraction is in its simplest form.

Fraction simplification is important because it makes numbers easier to work with and compare.

Multiplication in Algebra

Multiplication is a fundamental operation in algebra used to scale numbers, both in the numerator and the denominator of fractions. When handling fractions, it's crucial during the evaluation of expressions. In our case, the expression \( \frac{y}{2z} \) required multiplying the denominator.
First, recognize the components you need to multiply, such as in \( 2 \times 5 \) from the expression. This helps in finding the value of the denominator.

• Identify each factor to multiply.
• Use multiplication to compute the product.

Once calculated, you obtain the result \( \frac{3}{10} \). Understanding multiplication's role in algebra helps clarify how expressions interact and are reduced.