Question

Explain what it means to evaluate \(a\) variable expression.

Short answer

Evaluating a variable expression involves substituting the value of the variable into the expression and then performing the mathematical operations to get a final numerical value. For example, if the expression is 2x + 3 where x=5, replace x with 5 to get 2*5 + 3 = 13. Therefore, the expression is evaluated to 13.

Step-by-step solution

Understanding Variable Expressions

A variable expression is a combination of numbers, variables and mathematical operations. For example, an expression could be 2x + 3, where 'x' is a variable.

The Evaluation Process

To evaluate a variable expression means to replace the variable in the expression with a specific numeric value. For instance, if x = 5, the evaluation of the given expression 2x + 3 would be 2*5 + 3.

Performing the calculations

After substituting the value of the variable in the expression, perform the necessary arithmetic operations. In the given example, it would be 2*5 + 3 = 10 + 3 = 13. So, the expression 2x + 3 evaluates to 13 when the value of x is 5.

Key concepts

Variable Expressions

Variable expressions are fundamental to algebra and appear frequently in mathematical equations. These expressions consist of variables, numbers, and arithmetic operations. Think of a variable as a placeholder for a number that might change or that might not even be known at the time you're working with the expression.

For example, take the expression 4y - 7. Here, '4y' means '4 times y', and the entire expression represents a number that depends on the value of y. Understanding variable expressions is like having a recipe where certain ingredients can be swapped out—change the variable, and the final result changes.

Importance in Mathematics

Variable expressions are used to write formulas, define functions, and solve problems where specific values are unknown or variable. They allow us to generalize problems and solve them for many possible inputs.

Substitute Variables

To substitute a variable means to replace it with a numerical value. This is a critical step in evaluating an expression. Just as a baker might substitute almond milk for regular milk in a recipe, a mathematician or a student can substitute a number for a variable in an expression to find a specific result.

How to Substitute

Let's say we have the expression 3x + 4 and we want to substitute x with the number 2. The process involves two main steps: First, identify the variable to be replaced, in this case, x. Second, replace all instances of x with the number 2, yielding 3(2) + 4. This substitution clarifies the next operations to be performed.

Arithmetic Operations

Arithmetic operations are the bread and butter of basic mathematics. They consist of addition, subtraction, multiplication, and division. When we evaluate a variable expression after substituting variables with their specific values, we use these operations to find our final answer.

Order of Operations

Always remember the order of operations, often abbreviated as PEMDAS or BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction. This order dictates the sequence in which operations should be carried out. For instance, after substituting the variables, multiplication and division are performed before addition and subtraction. Applying this order to our substituted expression 3(2) + 4, we first do the multiplication to get 6 + 4, and then we add the numbers to arrive at the final answer, 10.