Problem 40 - College Algebra

Question

evaluate each algebraic expression for the given value of the variable or variables. $$ 9 x+6 ; x=5 $$

Step-by-Step Solution

Answer

The evaluated result for the expression \( 9x + 6 \) when \( x = 5 \) is 51.

Step 1: Identify the expression and the value of the variable

The algebraic expression to be evaluated is \( 9x + 6 \) and the given value of the variable \( x \) is 5.

Step 2: Substitute the value into the algebraic expression

Replace \( x \) in \( 9x + 6 \) with the value 5, which will give us \( 9(5) + 6 \).

Step 3: Evaluate the Expression

Using the multiplication and addition order of operations, i.e., performing multiplication before addition, the result will be \( 9*5 + 6 = 45 + 6 = 51 \).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). To understand them, think of algebraic expressions as recipes that tell us how to mix exactly the right ingredients (numbers and variables) to create a new mathematical dish. The beauty of these expressions is that they are not fixed; they change their value when we alter the variables within them.

For example, take the algebraic expression from the exercise, 9x + 6. Here, 9 and 6 are constants, while x is a variable which can take on different values. The constant 9 is called the coefficient of x, revealing how many times x is being multiplied. These elements are combined with the operation of addition which ultimately gives this expression its unique characteristics and potential results.

Variable Substitution

The concept of variable substitution is a bit like playing a game of 'fill in the blanks.' In algebra, when an expression contains a variable, you aren’t stuck with an abstract symbol; instead, you can replace that symbol with a real number to find out what the expression is worth. This is incredibly helpful when you want to evaluate the expression for specific values.

In the exercise, we replace the variable x in the expression 9x + 6 with the number 5. So the expression 9x becomes 9(5), which simplifies to 45 when you multiply the 9 by 5. By substituting and then simplifying, you're effectively decoding the numerical value hidden within the algebraic expression for that particular value of the variable.

Order of Operations

To accurately evaluate algebraic expressions, we rely on the order of operations, a universal set of rules that mathematicians agree upon to avoid chaos in calculations. The basic order to follow is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), which you might remember by the acronym PEMDAS.

When we apply the order of operations to the expression 9(5) + 6, we first do the multiplication 9(5) which equals 45. Then we perform the addition: 45 + 6 equals 51. It's essential to follow these steps in the correct sequence; otherwise, you may end up with the wrong answer. Think of it as a math recipe, where steps taken out of order can result in a very different—and possibly unpalatable—outcome.