Problem 18 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 21.

Step 1: Substitute Values into the Expression

First, substitute the given values of the variables into the expression. The expression is \( \frac{9x}{y} \). Substitute \( x = 7 \) and \( y = 3 \) into the expression, which gives: \( \frac{9 \times 7}{3} \).

Step 2: Calculate the Numerator

Multiply the coefficient 9 by the value of \( x \), which is 7. Thus, the expression becomes \( \frac{63}{3} \) after performing the multiplication: \( 9 \times 7 = 63 \).

Step 3: Simplify the Expression

Perform the division in the expression \( \frac{63}{3} \). Divide 63 by 3 to simplify: \( 63 \div 3 = 21 \). So the expression simplifies to 21.

Key Concepts

Understanding the Substitution Method

The substitution method is a powerful tool to solve expressions or equations by replacing variables with specific values. In the original exercise, we used the given values for variables like \( x = 7 \) and \( y = 3 \). When substituting, we plug these numbers directly into the expression where the variables appear. This transforms the algebraic expression into a simplified numerical one, which is easier to evaluate. The substitution method is helpful because it effectively allows us to work with numbers rather than abstract symbols, giving us concrete results to interpret.
When using substitution, always

• Identify the variables that need replacing.
• Carefully replace each instance of the variable in the expression with its given value.
• Check to ensure all substitutions maintain the integrity of the original expression.

Practice makes perfect, and the more you apply this method, the more intuitive it becomes, sharpening your problem-solving skills.

Navigating Numerical Expressions

Numerical expressions consist purely of numbers and operations, without any variables. Once you've substituted values for the variables, what remains is a straightforward numerical expression. For example, after substitution, the expression \( \frac{9 \times 7}{3} \) becomes a pure numerical expression like \( \frac{63}{3} \).
Understanding numerical expressions involves recognizing the order of operations. Follow the rules of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) for accurate results. Here:

• First, perform multiplication within the numerator.
• Next, execute division to arrive at a simplified result.

This understanding allows you to navigate complex problems by breaking them down into numerical parts that are simpler to handle.

Mastering Simplifying Expressions

Simplifying expressions is the final step to reaching a solution in exercises like the one given. It involves reducing an expression into its simplest form without changing its value. This makes expressions more manageable and the results more understandable.
In the exercise, after evaluating the multiplication, we had \( \frac{63}{3} \). To simplify, you perform the division \( 63 \div 3 \), resulting in 21. Here are a few tips to simplify expressions effectively:

• Ensure all arithmetic within parentheses is resolved first.
• Systematically carry out multiplication and division from left to right as they appear.
• Reduce any fraction to its simplest form, as seen in the division of 63 by 3.

This process not only gives you the final answer but enhances your understanding of how expressions build and simplify to convey clearer mathematical insights.