Problem 40 - Prealgebra

Question

Evaluate each expression if \(x=9, y=4,\) and \(z=12\) $$(29-3 y)+4 z-7$$

Step-by-Step Solution

Answer

The value of the expression is 58.

Step 1: Substitute the Variables

Start by substituting the given values into the expression. Replace every instance of \(x\) with 9, \(y\) with 4, and \(z\) with 12 in the expression. The expression becomes: \((29 - 3(4)) + 4(12) - 7\).

Step 2: Perform Multiplication

Perform the multiplication operations in the expression. Multiply \(3\) by \(4\) and \(4\) by \(12\). This results in:\(29 - 12 + 48 - 7\).

Step 3: Evaluate Additions and Subtractions

Now perform the additions and subtractions from left to right. Start with \(29 - 12\), which gives \(17\). Then add \(48\) to \(17\) which gives \(65\). Finally, subtract \(7\) from \(65\) to get \(58\).

Key Concepts

Variable Substitution

In prealgebra, one common task is replacing variables with numbers in expressions, which is known as variable substitution. This technique is vital because it transforms algebraic expressions into numeric ones that can be more easily evaluated.
To perform variable substitution, follow these steps:

• Identify the variables within the expression.
• Use the given values for each variable. For instance, if you are given that \(x=9\), \(y=4\), and \(z=12\), replace every \(x\) with 9, \(y\) with 4, and \(z\) with 12.
• Substitute these values into your expression, maintaining the original structure of the equation.

By following these steps, an expression like \((29-3y)+4z-7\) becomes \((29-3(4))+4(12)-7\). This conversion makes the expression ready for evaluation using standard arithmetic operations.

Order of Operations

After substituting variables, the next step is to solve the expression using the correct order of operations. In mathematics, there is an agreed-upon order that must be followed to ensure accurate results. This order is often remembered by the acronym PEMDAS:

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

In the expression \((29-3(4))+4(12)-7\), start with the operation within the parentheses by performing the multiplication \(3 \times 4\). Then, do the multiplication \(4 \times 12\).
After completing these multiplications, handle the remaining operations by performing additions and subtractions sequentially from left to right. This careful adherence to order guarantees that each calculation is performed correctly.

Arithmetic Expressions

Arithmetic expressions in prealgebra involve performing operations like addition, subtraction, multiplication, and, sometimes, division on numbers. These expressions can often include variables that need to be substituted with numerical values to solve them.
In the solved exercise, we dealt with the arithmetic expression \(29 - 12 + 48 - 7\) after substituting the variables and performing initial multiplications. Each number and operation stands as a part of a sequence, where:

• The numbers represent quantities or values.
• The operations among them dictate how these numbers interact.

By conducting calculations from left to right, starting with subtraction \(29-12\), then adding \(48\), and finally subtracting \(7\), you systematically reduce the expression to a single numerical outcome: \(58\).
Understanding how to evaluate such expressions is fundamental to grasping more complex algebraic concepts later on.