Question

Evaluate the expression for the given value of the variable. $$12 x+3 \text { when } x=5$$

Short answer

The evaluation of the expression \(12x+3\) when \(x=5\) is 63.

Step-by-step solution

Substitute

The first step is to substitute the given value for the variable x in the given expression. This would give \(12(5) + 3\).

Simplify

Now it's time to simplify the expression we obtained in step 1. Multiply 12 and 5 to get 60 and then add it to 3 to arrive at the final answer.

Key concepts

Substituting Variables

The process of substituting variables involves replacing the variables with their given numerical values. It's the fundamental step in evaluating expressions in algebra. Imagine that a variable is a placeholder for a number; when you know what number it represents, you can swap it out.

Let's consider an example where we have an algebraic expression like \(12x + 3\) and we're given that the value of \(x\) is 5. Substituting the variable means we replace every instance of \(x\) with 5, resulting in the new expression \(12(5) + 3\). It's the initial transformation that changes an abstract algebraic expression into a more concrete and solvable form. This step makes it possible to move forward with calculation to find the final value of the expression.

Simplifying Expressions

Once variables have been substituted, the next step is to simplify the expression. This involves performing arithmetic operations such as addition, subtraction, multiplication, and division according to the order of operations—remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In the continuing example, after substituting \(x\) with 5, we get the expression \(12(5) + 3\). Simplifying this expression requires us to multiply 12 by 5 first, and then add 3 to the product. So, it progresses as follows: \(12 \times 5 = 60\) and then \(60 + 3 = 63\). The simplified result of this expression would be 63. Simplifying expressions reduces them down to their simplest form, making it easier to understand the result.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. Unlike equations, they do not have an equal sign and thus do not signify a relationship of equality. Expressions can look daunting with letters and numbers jumbled together, but they follow specific rules and patterns.

For example, the expression \(12x + 3\) is made up of terms \(12x\) and \(+3\). Here, \(12x\) represents the variable term because it includes the variable \(x\) multiplied by a constant 12, and \(+3\) represents the constant term that stands alone without a variable. These are the building blocks of algebra, and by learning to work with them, students can move on to solve more complex problems with ease.

Basic Algebra

Basic algebra is where variables and numbers start interacting. It lays the foundation for all further mathematical study. Understanding the principles of basic algebra involves mastering how to manipulate expressions and solve equations.

Engaging with basic algebra starts with learning how to handle expressions by substituting values for variables and simplifying to find their numerical worth. For instance, taking our expression from before \(12x + 3\), with basic algebra skills, a student can easily identify that \(x\) is the variable, the numbers 12 and 3 are coefficients and constants respectively, and performing the necessary operations following the substitution of \(x\) with a given value leads to the solution.