Problem 22 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$\frac{x y}{3}+2$$

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Substitute the values of variables

Replace the variables in the expression with the given values: \(x = 7\), \(y = 3\), and \(z = 9\). The expression becomes \(\frac{7 \times 3}{3} + 2\).

Step 2: Perform the multiplication in the numerator

Calculate the product inside the fraction: \(7 \times 3 = 21\). The expression now simplifies to \(\frac{21}{3} + 2\).

Step 3: Simplify the fraction

Divide the numerator by the denominator: \(\frac{21}{3} = 7\). The expression is now \(7 + 2\).

Step 4: Solve the addition

Add the remaining numbers: \(7 + 2 = 9\). Thus, the value of the expression is 9.

Key Concepts

Substitution in Algebra

Substitution is like replacing ingredients in a recipe. You swap out letters for numbers. This helps solve or evaluate algebraic expressions where letters stand for unknown values. In the context of this problem, substitution means replacing the variables \(x, y,\) and \(z\) with their given values.

• Given: \(x = 7\), \(y = 3\), \(z = 9\).
• Expression: \(\frac{x y}{3} + 2\).
• Substitute values: The expression becomes \(\frac{7 \times 3}{3} + 2\).

The key is ensuring each substitution is correct. A mistake here means the final answer will be incorrect. Once we've substituted, the expression becomes a series of numbers we can calculate.

Understanding Algebraic Expressions

Algebraic expressions are like math sentences. They use numbers, variables, and operations to represent ideas or quantities. In this exercise, the expression is \(\frac{xy}{3} + 2\). Here are the basic components:

• Variables: \(x\) and \(y\) are placeholders for numbers.
• Operations: Multiplication, division, and addition are used.
• Constants: The numbers 3 and 2 are fixed values.

An algebraic expression does not include an equals sign; it cannot be solved for a specific value without additional information. Instead, expressions can be simplified or evaluated by substitution of known values, making them incredibly useful for math problems.

Simplification for Easy Solutions

Simplification helps us break down complicated expressions into simpler parts. By reducing an expression to its simplest form, calculations become straightforward. In this problem:1. **Multiplication in the Numerator**: Multiply the numbers in the fraction's top part: \(7 \times 3 = 21\).2. **Simplify the Fraction**: Divide the product by the number in the denominator: \(\frac{21}{3} = 7\).3. **Complete the Expression**: Add the remaining number: \(7 + 2 = 9\).Every step in simplification should be methodical. This ensures no errors occur during computation. It's like cleaning your workspace before starting a project, making it easier to see the task clearly and complete it accurately.