Problem 59 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \((L e s s o n \quad 1.1)\). $$\frac{24}{x} when x=3$$

Step-by-Step Solution

Answer

The value of the expression \(\frac{24}{x}\) when \(x=3\) is 8.

Step 1: Understand the Queries

The exercise asks to evaluate an expression \(\frac{24}{x}\) when \(x\) is 3. Evaluate is another term for solve.

Step 2: Substitute

Replace \(x\) with 3 in the given expression, resulting in \(\frac{24}{3}\).

Step 3: Perform Division

Perform the division to solve the expression: \(\frac{24}{3} = 8\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as addition, subtraction, multiplication, and division). To evaluate an algebraic expression, we replace each variable with a specific value and perform the arithmetic operations according to the established order of operations — parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.

For instance, if we have the expression \( \frac{2x}{5} + 3 \), evaluating it for \( x = 10 \) would require substituting \( x \) with 10, resulting in \( \frac{2 \times 10}{5} + 3 \), which simplifies to \( 4 + 3 \) and finally gives us a value of 7.

Substitution Method

The substitution method is a foundational technique used in algebra to solve equations or evaluate expressions. It involves replacing a variable in an expression with a given number. The beauty of substitution is in its simplicity—it allows us to turn an abstract algebraic expression into a more concrete numerical one that we can compute.

Let's take the expression \( 3y + 2 \) as an example. If we are told that \( y \) equals 4, we substitute the 4 in place of \( y \) to get \( 3 \times 4 + 2 \) which simplifies to \( 12 + 2 \) and gives us the final result of 14. By substituting step by step, complex problems become much more manageable.

Performing Division

Performing division, in its most basic form, is splitting a quantity into equal parts. In algebra, when we are provided with a fraction, like \( \frac{24}{x} \), which we encounter in expressions, performing division correctly is crucial.

To divide numeric expressions, we write them in fraction form and reduce them by finding a number that can divide both the numerator (the top number) and the denominator (the bottom number) without a remainder. For example, \( \frac{24}{3} \) simplifies to 8 because 24 divided by 3 equals 8. In cases where the expression cannot be simplified to a whole number, we may leave the answer in fraction or decimal form, ensuring we've performed the division correctly.