Question

Evaluate the expression for the given value of the variable. (Review 1.1) \(8 a\) when \(a=4\)

Short answer

After evaluating the given expression for \(a=4\), the result is 32

Step-by-step solution

Substitute Variable

Replace \(a\) in the expression with the given value 4. This gives us \(8 \times 4\).

Perform Multiplication

Now we have a simple multiplication to perform: \(8 \times 4 = 32\).

Key concepts

Substitution in Algebra

Understanding how to handle variables is essential in algebra, and substitution is a foundational technique used to evaluate algebraic expressions. At its core, substitution involves replacing variables, which are symbols that represent unknown quantities, with their corresponding values. For instance, consider the expression 8 * a. The variable here is a. If we are told that a = 4, substitution simply means putting 4 in place of a in the expression.

Here's how to do it step by step: First, identify the variable in the expression. Next, determine the value that the variable represents—this will be provided in the problem or by solving a related equation. Finally, replace the variable with the given value. In our example, this results in the new expression: 8 * 4. Accurate substitution ensures that the integrity of the expression is maintained while allowing us to move ahead with solving the problem.

Basic Multiplication

Multiplication is one of the four elementary mathematical operations that are fundamental to working with numbers. It can be thought of as repeated addition. For example, the expression 8 * 4 essentially means adding the number 8 to itself 4 times. A clear understanding of multiplication tables and the ability to perform basic multiplication quickly are invaluable when evaluating algebraic expressions.

After substituting the variable with its value, we often end up with a multiplication task, as we do in our example. Multiplying numbers is straightforward: align the numbers one under the other and multiply each digit of the top number by the lower number, adding the results. Modern students can also make good use of digital tools, like calculators, to ensure accuracy and efficiency in multiplication. Learning multiplication not only helps with algebra but is a skill that you'll use throughout all areas of math and daily life.

Variable Evaluation

Variable evaluation is the process of finding the value of an expression once all the variables have been substituted. This is a crucial step in solving algebraic expressions and requires a proper understanding of the order of operations, which dictates the sequence in which to perform arithmetic operations. In evaluating the variable, it's important to remember PEMDAS/BODMAS rules, which stand for Parentheses/Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (left-to-right), and Addition and Subtraction (left-to-right).

In our example, after substituting a with 4, we are left with the expression 8 * 4. We then follow through with the multiplication operation (since there are no parentheses or exponents to deal with), leading to 32. Hence the variable a is evaluated to result in the numeric solution. The ability to evaluate variables effectively is a key skill in algebra that enables problem-solving and helps in understanding more complex algebraic concepts.