Problem 59 - Beginning and Intermediate Algebra

Question

Evaluate each expression if \(x=12, y=8,\) and \(z=4.\) \(x^{2}-3 y+x\)

Step-by-Step Solution

Answer

The evaluated expression is 132.

Step 1: Substitute the Values for the Variables

Replace the variables in the expression with their given values. The expression is \(x^2 - 3y + x\). Substitute \(x = 12\) and \(y = 8\). So, it becomes \(12^2 - 3(8) + 12\).

Step 2: Calculate the Square

Compute the square of \(x\). For \(12^2\), calculate \(12 \times 12 = 144\). Now the expression is \(144 - 3(8) + 12\).

Step 3: Multiply and Simplify

Multiply \(3\) by \(8\). The result is \(24\). Substitute in the expression to get \(144 - 24 + 12\).

Step 4: Perform the Addition and Subtraction

Subtract \(24\) from \(144\) to get \(120\). Then add \(12\) to \(120\) to obtain \(132\).

Key Concepts

Substitution Method

In algebra, the substitution method helps simplify complex expressions by replacing variables with given numerical values. When evaluating expressions with specific values for variables, you use substitution to make calculations easier.Consider the expression given: \(x^2 - 3y + x\). By substituting the values \(x = 12\) and \(y = 8\), the expression transforms into \(12^2 - 3(8) + 12\). This step makes it possible to convert an algebraic expression into a purely numerical one.Using the substitution method effectively:- Identify all variables in the expression.- Replace each variable with its corresponding value given in the problem.- Simplify the expression as it becomes purely arithmetic.Substitution is a powerful tool in algebra, providing clarity and ease when working with expressions.

Order of Operations

Understanding the order of operations is crucial when evaluating any algebraic expression. This set of rules dictates the order in which calculations should be performed to achieve the correct result, commonly remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M/D: Multiplication and Division (left-to-right)
• A/S: Addition and Subtraction (left-to-right)

In our example, the expression \(12^2 - 3(8) + 12\) should be simplified by following these rules. First, we calculate the exponent \(12^2\):- Exponents have higher priority, so we compute \(12 \times 12 = 144\).Next come multiplication operations, like \(3 \times 8\):- Perform multiplication before addition and subtraction, giving \(24\).Finally, solve addition and subtraction from left to right:- Start with \(144 - 24\), resulting in \(120\).- Then, add \(12\): \(120 + 12 = 132\).Following the correct order is essential to ensure accurate results in math problems.

Arithmetic Operations

Arithmetic operations form the foundation of evaluating expressions, comprising the basic functions of addition, subtraction, multiplication, and division. Each of these operations plays a specific role in simplifying algebraic expressions.Consider our exercise where the expression \(12^2 - 3(8) + 12\) requires these operations:- **Multiplication**: Before anything else, the expression includes multiplication, as seen in \(3 \times 8\), which is simplified to \(24\).- **Addition and Subtraction**: These two are performed after exponents and multiplication, starting from left to right. First subtract \(24\) from \(144\), resulting in \(120\), and finally adding \(12\) to get \(132\).Understanding each operation's role and proper application makes algebraic expressions easier to manage:- **Multiplication and division** can alter the magnitude of numbers and are solved before tackling addition and subtraction.- **Addition and subtraction** help refine and consolidate the final output.Mastering these operations ensures clarity and precision when handling mathematical expressions.