Problem 31 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ \frac{18}{x} \text { when } x=3 $$

Step-by-Step Solution

Answer

The value of the expression \(\frac{18}{x}\) when \(x=3\) is 6

Step 1: Substitute the given value

Replace \(x\) in the expression \(\frac{18}{x}\) by 3. This results in the expression becoming \(\frac{18}{3}\).

Step 2: Perform the Division

Perform the operation of division which is \(\frac{18}{3}\). The result is 6.

Key Concepts

Substitution Method

The substitution method in mathematics is a technique used to evaluate expressions by replacing variables with their given values. This method is fundamental when dealing with algebraic expressions, as it allows us to find the value of an expression for particular variables.

Take, for instance, the expression \(\frac{18}{x}\) from the problem. When we are told that \(x=3\), we substitute 3 for \(x\) in the expression. The process of substituting involves a very direct replacement, converting \(\frac{18}{x}\) into \(\frac{18}{3}\). This simplified form of the expression is now ready for us to perform the arithmetic operation—in this case, division.

Why Use the Substitution Method?

Using the substitution method allows students to see how the value of an expression changes as the variables within change. It's a powerful tool for understanding the behavior of mathematical models in the real world and is especially relevant when working through problems that involve formulas or functions.

Algebraic Expressions

Algebraic expressions are combinations of letters (variables) and numbers often linked by the operations of addition, subtraction, multiplication, and division. Unlike an equation, an algebraic expression doesn't have an equality sign—think of it more like a phrase rather than a full sentence.

In our example, \(\frac{18}{x}\) is an algebraic expression where 18 is a constant and \(x\) is a variable. It's algebraic because it involves a variable, and it's an expression because it's not equated to anything else.

Understanding through Evaluation

Evaluating algebraic expressions by plugging in values, as we did with \(x=3\), shows how mathematics can model situations flexibly. When a student learns to evaluate expressions, they're developing not just computational skills but also a deeper appreciation for the abstract nature of algebra.

Performing Division

Division is one of the basic arithmetic operations and is the process of determining how many times one number is contained within another. When performing division with algebraic expressions, we follow the same rules as with numerical division.

In the exercise, once we substituted the variable \(x\) with the value 3, we were left with \(\frac{18}{3}\). To perform this division, we divide the numerator (18) by the denominator (3), which gives us the result of 6. This is a straightforward numerical operation, but it's crucial to confirm that the denominator is not zero, as division by zero is undefined in mathematics.

Division as an Essential Skill

Mastering division is essential for higher-level math. In algebra, understanding how to manipulate and divide expressions is vital for simplifying expressions, solving equations, and grasping more advanced topics like rational expressions.