Problem 4 - College Algebra Essentials

Question

Evaluate each algebraic expression for the given value or values of the variable(s). $$8 x-y, for\quad x=3\quad and\quad y=4$$

Step-by-Step Solution

Answer

The simplified value of the algebraic expression for the given values \(x = 3\) and \(y = 4\) is 20.

Step 1: Identify Variables and Their Given Values

In the given algebraic expression \(8x - y\), \(x\) and \(y\) are the variables. From the problem statement, the given values for the variables are \(x = 3\) and \(y = 4\).

Step 2: Substitute Variables with Their Values

Substitute the given values of the variables into the algebraic expression. This gives \(8*3 - 4\).

Step 3: Simplify the Expression

Perform the arithmetic operations in the expression to simplify it into a single value. The multiplication operation is done first according to the rules of order of operations (BIDMAS/BODMAS, or PEMDAS in US). This results in \(24 - 4\). Next, subtract 4 from 24 to get the final result.

Key Concepts

Substitution in Algebra

Understanding the concept of substitution in algebra is essential for evaluating expressions accurately. Substitution is the process where we replace variables with their given numeric values. In the exercise given, you have the expression 8x - y, where x and y are variables representing numbers. Substituting variables is like stepping into the shoes of that variable with a numerical value.

For instance, when we are told that x = 3 and y = 4, we literally place these numbers in the expression wherever we see the corresponding variables. Doing this gives us a new expression 8 * 3 - 4, which no longer contains any variables, just concrete numbers we can work with. The act of substitution is the first step toward finding the value of the expression, paving the way for simplification.

Simplifying Expressions

Once substitution is done, the next step is to simplify the expression. Simplifying means performing the arithmetic to condense the expression into the most reduced form or a single number. This includes carrying out addition, subtraction, multiplication, and division as per the presence in the expression.

In our case, after substitution we get 8 * 3 - 4, and simplifying this involves executing the multiplication resulting in 24 - 4. Following this, we perform the subtraction operation, which simplifies the expression further to just 20. Always remember, simplifying algebraic expressions is not about changing the expression's value; it's about making it easier to understand and interpret by reducing it to the simplest numerical form.

Order of Operations

The order of operations is a fundamental concept to ensure that mathematical expressions are interpreted uniformly. To avoid ambiguity, mathematicians have agreed upon a specific order in which arithmetic operations should be performed. This order is often memorized using acronyms such as BIDMAS/BODMAS in the UK (Brackets, Indices/Order, Division and Multiplication, Addition and Subtraction) or PEMDAS in the US (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

This hierarchy instructs that in a single expression, operations in parentheses or brackets are carried out first, followed by exponents or indices, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). The expression from our exercise, after substitution, is 24 - 4. This is straightforward, as we only have one operation left: subtraction. Hence, we just subtract 4 from 24, resulting in 20. Remember, always adhere to the prescribed order to arrive at the correct result.