Problem 73 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(\frac{24}{s}\) when \(s=8\)

Step-by-Step Solution

Answer

The evaluation of the expression \(\frac{24}{s}\) for \(s=8\) is 3.

Step 1: Substitute the Given Variable

Replace the variable 's' in the expression \(\frac{24}{s}\) with the given value 8. This changes the expression into \(\frac{24}{8}\).

Step 2: Evaluate the Expression

Now perform the division operation: \(\frac{24}{8} = 3\). Therefore, the value of the expression \(\frac{24}{s}\) when \(s = 8\) is 3.

Key Concepts

Substitution in Algebra

Substitution is a foundational technique in algebra where we replace variables with their given numerical values. The process is quite straightforward but critical for accurately evaluating algebraic expressions. Consider the problem at hand, where we needed to evaluate \( \frac{24}{s} \) for \( s=8 \) . Here's how we applied substitution effectively:
First, identify the variable in your expression. In this case, it's the letter \( s \) in the denominator. Next, take the given value of the variable, which is 8, and replace the variable with this number. Thus, substituting the \( s \) in the expression with 8 transforms \( \frac{24}{s} \) into \( \frac{24}{8} \) . This direct replacement sets the stage for simplifying the expression in the next step.

Simplifying Expressions

Once substitution has been completed, the next task in evaluating algebraic expressions is simplifying them. Simplification makes the expression easier to understand and solve. To simplify an expression, we apply the appropriate algebraic operations like addition, subtraction, multiplication, and division, as they apply to the given numbers. In our exercise, after the substitution step, we are left with \( \frac{24}{8} \) which is a straightforward division problem. To simplify \( \frac{24}{8} \) , we divide 24 by 8 yielding the simple, reduced form of 3. This simplification step reveals the evaluated result of the expression for the provided variable value.

Algebraic Operations

Handling algebraic operations is essential in evaluating expressions. The four basic operations are addition, subtraction, multiplication, and division. Each operation has its specific rules and applications. For example, when multiplying or dividing terms with the same base, we can add or subtract their exponents due to the laws of exponents. It's important to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Always apply operations inside parentheses first, and perform multiplication and division before addition and subtraction, unless grouping symbols indicate otherwise. Proper grasp and application of these operations enable us to solve algebraic expressions effectively.

Division Operation in Algebra

Division is one of the critical algebraic operations and is often one where students may stumble, particularly with variables involved. In algebra, division is essentially the multiplication of a number by the reciprocal of another number. Thus, understanding how to perform division with variables is key. In our example, we divided the constant 24 by the variable's specified value, which was 8. While working with division in algebra, especially with variables, it's important to ensure that the divisor (the number by which you divide) is not zero, as division by zero is undefined in mathematics. In expressions that become more complex, always reduce terms where possible to make the division as straightforward as possible, and be mindful of simplifying fractions to their lowest terms.