Question

Evaluate the expression for the given value of the variable. $$b+6 \div 4 \text { when } b=1.5$$

Short answer

The result of the expression when \(b=1.5\) is 3.

Step-by-step solution

Substitution

Replace \(b\) in the expression with the given value 1.5. So the expression becomes \(1.5 + 6 \div 4\).

Perform Division

According to the order of operations, carry out the division before addition. That is \(6 \div 4 = 1.5\), so the expression becomes \(1.5 + 1.5\).

Perform Addition

Now, perform the addition. Hence, \(1.5 + 1.5 = 3\).

Key concepts

Order of Operations

Understanding the order of operations is crucial in algebra for evaluating expressions correctly. Often summarized by the acronym PEMDAS, it stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This sequence determines which part of the expression should be calculated first to ensure everyone gets the same answer.

For example, let's apply the order of operations to the expression from the exercise: when asked to evaluate \(b+6 \div 4\) with \(b=1.5\), notice how division comes before addition in PEMDAS. Therefore, you divide 6 by 4 first before adding the result to 1.5. By following these rules, we avoid any confusion that might arise from performing operations out of order and ensure the expression is evaluated accurately.

Substitution in Algebra

The concept of substitution in algebra involves replacing variables with their given values to simplify expressions and solve equations. It's a straightforward process but is foundational to correctly evaluating algebraic expressions.

In the given problem, we see substitution at work when the letter 'b' is replaced with its given value (1.5). After substitution, you have concrete numbers to work with, making it possible to apply arithmetic operations. This process is the first step in the exercise and is pivotal because it turns an abstract expression into a calculable numerical one. Substitution is used in various fields of math and is a stepping stone to understanding more complex algebraic concepts.

Arithmetic Operations

The backbone of algebra, arithmetic operations include addition, subtraction, multiplication, and division. They're the most basic yet essential elements of math, used to evaluate expressions and solve equations.

In our example, after substituting 1.5 for 'b', we're left with two operations: division and addition. We divide 6 by 4 to get 1.5 and then add that to our substituted value for 'b'. It's the correct use of arithmetic operations, in conjunction with the order of operations, that gets us to the final answer, which in the exercise is 3. With a solid grasp of these operations, students can solve a vast array of mathematical problems, as they are not only used in algebra but in almost all mathematical calculations.