Problem 38 - Prealgebra

Question

Evaluate each expression if \(x=9, y=4,\) and \(z=12\) $$6 x-(z-2 y)$$

Step-by-Step Solution

Answer

The evaluated expression is 50.

Step 1: Substitute Given Values

First, replace the variables in the expression with the given values. The expression is \(6x - (z - 2y)\). Substitute \(x = 9\), \(y = 4\), and \(z = 12\) into the expression: \[6(9) - (12 - 2(4))\].

Step 2: Calculate Inside Parentheses

Next, solve the expression inside the parentheses. Start with \(2(4)\): \(2 \times 4 = 8\). Then subtract this from \(z = 12\): \(12 - 8 = 4\). So, the expression becomes \[6(9) - 4\].

Step 3: Calculate Multiplication

Now calculate the multiplication \(6(9)\): \(6 \times 9 = 54\). Thus, the expression simplifies to \[54 - 4\].

Step 4: Perform Final Subtraction

Finally, subtract \(4\) from \(54\): \(54 - 4 = 50\).

Key Concepts

Expression Evaluation

Expression evaluation is a crucial concept in mathematics, especially when working with algebra. It involves finding the value of an algebraic expression by replacing variables with their given values and then simplifying the expression fully. For example, in the expression \(6x - (z - 2y)\), we evaluate it by substituting specific numbers for variables: \(x = 9, y = 4,\) and \(z = 12\). By doing this, the expression becomes \(6(9) - (12 - 2(4))\). The primary goal of expression evaluation is to simplify this until a single numerical value remains. To accomplish this, it's essential to perform a series of arithmetic operations as defined by the rules of the problem, ensuring accuracy and correctness in the solution.

Substitution Method

The substitution method is a fundamental technique in algebra where you replace variables in an expression with specific given numbers. This transforms an abstract algebraic problem into a straightforward numerical problem.Here's how it works: Begin with an expression like \(6x - (z - 2y)\). Next, take the given values of the variables—\(x = 9\), \(y = 4\), \(z = 12\)— and substitute them into the expression wherever you see those variables.After making these substitutions, the expression becomes \(6(9) - (12 - 2(4))\). It's akin to filling in the blanks, which simplifies the expression and makes it easy to perform subsequent arithmetic operations.

Order of Operations

The order of operations is a set of rules that dictate the correct order to solve different parts of a mathematical expression. It often goes by the acronym PEMDAS:- **P**arentheses first- **E**xponents (or powers and roots, etc.) - **MD** Multiplication and Division (left to right) - **AS** Addition and Subtraction (left to right)When evaluating expressions, following these rules is crucial to get the correct result. For instance, consider the expression \(6(9) - (12 - 2(4))\). According to the order of operations, you start by addressing the inside of the parentheses first: calculate \(2(4)\), then subtract that result from 12.Once the parentheses are resolved, proceed by multiplying \(6\) by \(9\). Finally, perform the subtraction at the end. This systematic approach ensures that expressions are simplified accurately, maintaining consistency in algebraic solutions.