Question

Evaluate the expression for the given value of the variable. $$\frac{22}{x} \div 2+16 \text { when } x=11$$

Short answer

The value of the expression when \(x = 11\) is 17.

Step-by-step solution

Substitute the value

Start by substituting the value of \(x\) in the given expression. This leads to: \(\frac{22}{11} \div 2 + 16\)

Perform the division

Next, perform the division operations from left to right. This gives: \(2 \div 2 + 16 = 1 + 16\)

Perform addition

Finally, perform the addition. This gives: \(1 + 16 = 17\)

Key concepts

Substitution Method

Understanding the substitution method is crucial when evaluating algebraic expressions, as it allows you to replace variables with their given values. In the context of the provided exercise, we apply this method by taking the value of the variable 'x', which is 11, and substituting it back into the expression. This action simplifies the expression to an arithmetic problem that can be solved using basic operations.

For example, given the expression \(\frac{22}{x} \) and the value of 'x' being 11, we substitute 11 in place of 'x' to get \(\frac{22}{11} \). It's like replacing a placeholder with its actual value, transforming an abstract expression into a concrete numerical one which we can calculate. This step is the foundation upon which the remaining steps are built.

Order of Operations

The order of operations is a set of rules that determines the sequence in which operations should be carried out to correctly evaluate an expression. It is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the exercise provided, once the substitution method has been applied, we must carefully use the order of operations to arrive at the correct answer. After substituting the value of 'x', the expression involves division and addition. According to the order of operations, we handle the division (\(\frac{22}{11} \div 2\)) before moving on to addition (adding 16 to the result of the division). Failure to follow these rules could lead to the wrong answer, which emphasizes the importance of understanding and applying these fundamental guidelines in every calculation.

Algebraic Division

Algebraic division is a process where you divide algebraic expressions by numbers or other expressions. In the case study given, after substituting the variable 'x' with 11, we carry out algebraic division. Because division is an inverse operation to multiplication, the step \(\frac{22}{11} \) simplifies to 2 by canceling out the matching numerator and denominator. Then, we proceed with dividing 2 by 2.

It’s crucial to remember to divide coefficients and to apply the laws of exponents when dividing terms with variables, when necessary. In more complex algebraic expressions, terms may need to be factored or simplified before division can be completed. While this particular exercise doesn't require those steps, familiarity with algebraic division paves the way for tackling more challenging algebraic problems in the future.