Problem 17 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$\frac{6 y}{z}$$

Step-by-Step Solution

Answer

The expression evaluates to 2.

Step 1: Substitute the Known Values

Start by substituting the given values of the variables into the expression. Replace \(y\) with 3 and \(z\) with 9 in the expression \(\frac{6y}{z}\). This gives us: \(\frac{6(3)}{9}\).

Step 2: Simplify the Expression

Next, simplify the expression by calculating the product in the numerator. Multiply 6 by 3 to get 18. The expression becomes \(\frac{18}{9}\).

Step 3: Perform the Division

Finally, divide 18 by 9 to evaluate the expression. The result of \(\frac{18}{9}\) is 2.

Key Concepts

Substitution Method

The substitution method is a straightforward technique used in algebra to replace variables with given values. This helps in simplifying and evaluating expressions more easily. When practicing substitution:

• Identify the variables in the expression. In our example, we have two variables: \(y\) and \(z\).
• Replace each variable with its provided value. For \(y = 3\) and \(z = 9\), substitute these values into the expression \(\frac{6y}{z}\).
• This substitution transforms our expression into \(\frac{6(3)}{9}\).

Substitution is essential because it reduces complex expressions to numerical calculations. With this method, you can directly evaluate expressions without ambiguity.

Simplifying Fractions

Simplifying fractions is an important step to make an expression easier to understand and solve. In our example, the expression after substitution is \(\frac{18}{9}\). Here's how you can simplify it:

• Begin by determining the greatest common divisor (GCD) of the numerator and the denominator. For 18 and 9, the GCD is 9.
• Divide both the numerator and the denominator by their GCD, which results in \(\frac{18 \div 9}{9 \div 9}\).
• Performing this division simplifies the expression to \(\frac{2}{1}\) or just 2.

Simplifying not only makes fractions more manageable but also reveals the final answer in the simplest form. It’s particularly useful in algebra where expressions can quickly become complicated without simplification.

Multiplication in Algebra

Multiplication is a fundamental operation in algebra that helps to simplify expressions, especially when combining terms with constants and variables. In our original problem, we see multiplication as part of the substitution process:

• After substituting the values, the expression \(\frac{6(3)}{9}\) involves multiplying 6 by 3 in the numerator.
• This operation results in a product of 18. This step is crucial before you can move forward to simplify the fraction.

Multiplication helps to condense expressions by combining similar terms or numbers, making it easier to manage subsequent operations like division or addition. Understanding multiplication in algebra is vital for accurately simplifying and evaluating expressions.