Problem 2 - Algebra and Trigonometry

Question

Evaluate each algebraic expression for the given value or values of the variable(s). $$ 8+6 x, \text { for } x=5 $$

Step-by-Step Solution

Answer

The value of the expression when \(x = 5\) is \(38\).

Step 1: Understand the Problem

We are given an algebraic expression, \(8 + 6x\), and we are asked to evaluate it for \(x = 5\). This means that wherever we see \(x\) in the algebraic expression, we will replace it with the number 5.

Step 2: Substitute the value

The algebraic expression is \(8 + 6x\). Now substitute \(x = 5\) in given expression, it becomes \(8 + 6*5\).

Step 3: Execute the Operation

Now that we've substituted, all that's left is to perform the operation. This entails conducting the arithmetic operation: \(8 + 6*5 = 38\).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators. Variables are used in the expression as placeholders for values that can vary or need to be solved for. For example, in the expression \(8 + 6x\), \(8\) is a constant, \(x\) is a variable, and \(+\) is an arithmetic operator indicating addition.

To understand algebraic expressions, it is crucial to identify the components and comprehend how they come together. The variable \(x\) stands for any number, and the expression can represent different values depending on the number substituted for \(x\). Let's say if \(x=1\), then the expression simplifies to \(8 + 6\times1 = 14\). If we change the value of \(x\), the outcome will also change.

Substitution Method

The substitution method is a technique used to evaluate expressions by replacing variables with their given numeric values. Understanding how to apply this method is essential to mastering algebra. To properly execute the substitution, follow each step carefully.

For instance, in our exercise, we used the substitution method to replace \(x\) with the number \(5\). The correct application of this method is vital to arriving at the correct solution: \(8 + 6\times5\). Missteps like forgetting to substitute every instance of the variable can lead to incorrect results. It's also important to maintain the order of operations after substituting, handling multiplication before addition in this case.

Arithmetic Operations

Arithmetic operations form the backbone of evaluating algebraic expressions, which include addition, subtraction, multiplication, and division. When an expression is ready to be simplified post-substitution, you carry out these operations according to the order of operations, or PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

In the example \(8 + 6\times5\), since there are no parentheses or exponents, we proceed with multiplication before addition. This results in \(8 + 30 = 38\), completing the arithmetic necessary to evaluate the expression. Grasping arithmetic operations is crucial as they are the tools that allow us to simplify and solve mathematical expressions.