Problem 8 - Intermediate Algebra

Question

Find the value of each algebraic expression at the given replacement values. See Examples 1 and 2 \(2 a-b\) when \(a=12\) and \(b=7\)

Step-by-Step Solution

Answer

The value of the expression is 17.

Step 1: Substitute Values into the Expression

Start by substituting the given values of the variables into the algebraic expression. Replace \(a\) with 12 and \(b\) with 7 in the expression \(2a - b\). This gives us \(2(12) - 7\).

Step 2: Evaluate Multiplication

Next, evaluate the multiplication in the expression. Calculate \(2 \times 12\) to get 24. The expression now becomes \(24 - 7\).

Step 3: Evaluate Subtraction

Finally, perform the subtraction in the expression. Subtract 7 from 24 to get 17. Thus, the value of the expression \(2a - b\) when \(a=12\) and \(b=7\) is 17.

Key Concepts

Substitution: A Core Step in Algebra

Substitution is a fundamental concept in algebra that involves replacing variables in an algebraic expression with given specific values. It allows us to find a numerical solution to an expression. Imagine that you have an expression like \( 2a - b \), where \(a\) and \(b\) are placeholders for unknown numbers. Substitution involves taking the values assigned to these placeholders and inserting them into the expression.

• Identify the variables in the expression.
• Insert the known values in place of the variables.
• Simplify the expression step-by-step using these values.

When substituting, ensure accuracy by double-checking that the right values are plugged into the correct variables. In our example, we replaced \(a\) with \(12\) and \(b\) with \(7\), converting our expression into \(2(12) - 7\). This starts the process of solving the expression numerically.

Evaluation: Simplifying with Care

Evaluation is the process of simplifying an expression to reach a numerical value after substitution has been done. Once substitution has taken place, you usually have a series of operations left to perform. Each operation should be handled carefully and in the correct order. In the expression \(2(12) - 7\):

• First, calculate the multiplication: \(2 \times 12 = 24\).
• Next, address any other operations such as addition or subtraction. Here, subtract \(7\) from \(24\).
• This results in \(24 - 7 = 17\).

Always follow the order of operations, often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure you evaluate expressions accurately. By evaluating step by step, you prevent mistakes that could arise from rushing through complex calculations.

Variable Replacement: Understanding the Process

Variable replacement is a straightforward yet essential technique in algebra. It refers to substituting variables in an expression with their given numerical values to make the expression easier to handle. This process is key when transitioning from a general form to a specific solution:

• Identify the variables present in the expression.
• Use the given problem to find the values you need to replace each variable with.
• Substitute these values into the expression to transform abstract algebra into solvable arithmetic.

In the original problem, the expression \(2a - b\) becomes \(2(12) - 7\) through variable replacement. This step turns an algebraic question into a simple numerical exercise, paving the way to evaluate and solve for the final numeric answer. Keeping track of each variable's value is vital for accuracy, ensuring each replacement is correct.