Problem 21 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$10-\frac{x z}{9}$$

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Identify the Expression and Substitutions

The expression to evaluate is \(10-\frac{x z}{9}\), where we need to substitute \(x=7\), \(y=3\), and \(z=9\). In this expression, only \(x\) and \(z\) are used.

Step 2: Substitute the Values

Replace \(x\) with 7 and \(z\) with 9 in the expression. This yields the new expression: \(10 - \frac{7 \cdot 9}{9}\).

Step 3: Simplify the Fraction

Calculate the product in the fraction: \(7 \cdot 9 = 63\). Therefore, our expression becomes \(10 - \frac{63}{9}\).

Step 4: Divide the Fraction

Perform the division: \(\frac{63}{9} = 7\). Now, the expression is simplified to \(10 - 7\).

Step 5: Calculate the Result

Subtract 7 from 10 to find the final result: \(10 - 7 = 3\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition or subtraction. Unlike numerical expressions, which contain only numbers and operations, algebraic expressions can include variables that represent unknown or varying values.
For instance, in the expression \(10 - \frac{x \cdot z}{9}\), the variables \(x\) and \(z\) take on specific values for different scenarios, allowing us to evaluate the expression under various conditions.
Understanding how to work with these expressions is a fundamental part of algebra that leads to solving equations and understanding more complex mathematical concepts.

• Variables can stand for any number, so evaluating an algebraic expression means calculating its value given specific numbers for these variables.
• Operations include adding, subtracting, multiplying, and dividing the numbers and variables involved.
• It is essential to follow the order of operations (parentheses, exponents, multiplication/division, addition/subtraction or PEMDAS) when simplifying or evaluating algebraic expressions.

The Substitution Method

The substitution method is a technique used to evaluate algebraic expressions by replacing the variables with given numbers. This process transforms the expression into a numerical one, making it easier to calculate.
For example, to evaluate the expression \(10 - \frac{x \cdot z}{9}\) when \(x=7\) and \(z=9\):

• First, substitute \(x = 7\) and \(z = 9\) into the expression.
• The expression becomes \(10 - \frac{7 \cdot 9}{9}\).

This step ensures that the calculation becomes straightforward, as it converts the expression from one that requires variable knowledge to one that involves only numerical arithmetic.
Employing substitution effectively requires accurately following the given variable values and ensuring all necessary operations are carried out according to the mathematical rules.

Prealgebra Fundamentals

Prealgebra is the branch of mathematics that lays the groundwork for algebra and includes basic concepts necessary for success in further math courses. It might involve:

• Understanding operations with whole numbers, fractions, and decimals.
• Recognizing and working with variables and expressions.
• Learning the order of operations, crucial for solving both numerical and algebraic expressions correctly.

The expression evaluation, as demonstrated in our exercise, illustrates many prealgebra fundamentals. Using the order of operations to handle the expression \(10 - \frac{7 \cdot 9}{9}\) demonstrates the transition from numbers to expressions incorporating variables and the importance of precise computation.
Prealgebra thus functions as a bridge, leading students smoothly from arithmetic to algebra by introducing concepts incrementally and reinforcing them through practice.