Problem 95 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). \(2 a-7\) when \(a=6\)

Step-by-Step Solution

Answer

The evaluated expression when \(a=6\) is 5.

Step 1: Identify the expression and the given value

Identify the algebraic expression, which in this case is \(2 a-7\), and the given value for the variable \(a\) which is 6.

Step 2: Substitution

Substitute the given value of the variable into the algebraic expression. This means replacing \(a\) with 6 in the expression, getting the expression \(2*6-7\).

Step 3: Simplify the expression

Simplify the expression following the order of operations. Multiplication is performed before subtraction, so first perform the multiplication 2*6 to get 12, and then subtract 7 from 12 to get 5.

Key Concepts

Substitution Method

Understanding how to evaluate algebraic expressions is crucial for students who aim to grasp the fundamentals of algebra. The substitution method plays a central role in this process. It involves taking an algebraic expression and replacing the variables with specific numerical values.

Imagine an algebraic expression as a recipe, and the variables as ingredients. When we know the specific quantity (numerical value) of an ingredient, we can substitute it into the recipe (expression) to create the final product (evaluated answer). For instance, in the expression given in the exercise, \(2a - 7\), we substitute the variable \(a\) with the value 6. This replacement leads to a new expression, \(2 \times 6 - 7\), which we can further simplify to find the result.

Order of Operations

Once substitution is complete, it's not enough to just perform any operation at random. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the correct sequence to follow when simplifying expressions.

Prioritizing operations in the correct order is similar to following the steps of a dance routine – each move comes at a specific time for the routine to make sense. In our expression after substitution \(2 \times 6 - 7\), we first carry out the multiplication (\(2 \times 6 = 12\)) before the subtraction, because multiplication comes before subtraction in our PEMDAS sequence. This ensures we accurately streamline the expression to its simplest form.

Simplifying Expressions

The final act in evaluating algebraic expressions is simplifying expressions. This is where you take the expression, which now only has numbers and operations after substitution, and 'simplify' it down to a single numerical answer.

To continue with our previous example, having followed the order of operations, we now carry out the arithmetic to simplify \(2 \times 6 - 7\) into \(12 - 7\), which equals 5. Simplifying expressions can sometimes involve combining like terms, distributing values, or factoring, but in this case, it's a straightforward arithmetic problem. Remember, simplification is about making complex, lengthy expressions more concise and easier to understand, much like summarizing a long story into a brief summary that gets to the point.