Problem 61 - Algebra A Combined Function

Question

Evaluate each expression when \(x=12, y=8,\) and \(z=4\). $$ \frac{x}{z}+3 y $$

Step-by-Step Solution

Answer

27

Step 1: Substitute Values

First, substitute the given values of the variables into the expression. We have \( x=12 \), \( y=8 \), and \( z=4 \). The expression is \( \frac{x}{z} + 3y \). After substitution, it becomes \( \frac{12}{4} + 3 \times 8 \).

Step 2: Simplify Division

Next, simplify the division operation. Calculate \( \frac{12}{4} \) which equals 3. Now the expression is \( 3 + 3 \times 8 \).

Step 3: Simplify Multiplication

Now, perform the multiplication. Calculate \( 3 \times 8 \) which equals 24. The expression has simplified to \( 3 + 24 \).

Step 4: Final Addition

Finally, perform the addition. Add 3 and 24 to get 27. Thus, the value of the expression is 27.

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing variables with their corresponding numerical values to solve an expression. It's like solving a puzzle where each piece represents a number, and you just need to fill in the placeholders (variables) with the correct pieces (values). In our problem, we substitute the values \( x = 12 \), \( y = 8 \), and \( z = 4 \) into the expression \( \frac{x}{z} + 3y \).

• Substitution transforms the expression into numbers instead of letters. This makes it easier to perform calculations.
• It's essential to ensure you substitute the correct values for each variable. Double-check to avoid mistakes.
• After substitution, the expression becomes \( \frac{12}{4} + 3 \times 8 \).

Understanding substitution is a foundational skill in algebra, used extensively in solving equations and simplifying expressions. Once you master it, you can tackle more complex problems with confidence.

Order of Operations

The order of operations is a crucial rule set in mathematics that clarifies which calculations should be done first in expressions. It's often remembered by the acronym PEMDAS:

• Parentheses – Solve anything inside parentheses first.
• Exponents – Next, evaluate exponents (or powers).
• Multiplication and Division – These come next, performed from left to right as they appear.
• Addition and Subtraction – Lastly, do these operations from left to right.

In our expression \( \frac{12}{4} + 3 \times 8 \), we don't have parentheses or exponents, so we move on to multiplication and division:
* First, handle \( \frac{12}{4} \) to get 3.
* Next, perform the multiplication \( 3 \times 8 \) which gives you 24.
After dealing with these, you finish with addition: \( 3 + 24 \).
Understanding and following the order of operations ensures that you solve expressions accurately, giving you the correct result every time.

Simplification in Algebra

Simplification in algebra is the process of reducing expressions to their most concise form while maintaining the same value. It's akin to tidying up your workspace; everything becomes clearer and more organized.
The steps for simplification include:

• Perform operations like division and multiplication first, according to the order of operations.
• Simplify each part of the expression step-by-step.
• Keep the expression in an easily understandable form, ensuring no unnecessary complexity remains.

In the given problem, after substituting the values, we simplify in several steps:
1. Calculate division \( \frac{12}{4} = 3 \).
2. Do the multiplication \( 3 \times 8 = 24 \).
3. Finally, add 3 and 24 to simplify the expression to 27.
Through simplification, you're not just finding the final solution; you're understanding each part of the expression. This practice makes mathematics more intuitive and aids in verifying your work for accuracy.