Problem 15 - Prealgebra

Question

Evaluate each expression if \(x=7, y=3,\) and \(z=9\) $$2+4 z$$

Step-by-Step Solution

Answer

The expression evaluates to 38.

Step 1: Substitute the Values

The expression given is \( 2 + 4z \). We need to substitute the given value of \( z \), which is \( z = 9 \), into the equation.

Step 2: Perform the Multiplication

Once the substitution is done, we have the expression \( 2 + 4 \times 9 \). First, calculate \( 4 \times 9 \) which equals \( 36 \).

Step 3: Perform the Addition

With the multiplication done, the expression simplifies to \( 2 + 36 \). Perform the addition to get \( 38 \).

Key Concepts

Substitution

Substitution is a foundational method in algebra used to simplify expressions by replacing variables with their known values. Imagine having an expression that involves variables like \(x\), \(y\), and \(z\), and you know their values. Substitution allows you to replace each variable with its corresponding number, which makes the expression easier to work on and solve.

In our exercise, we're given the expression \(2 + 4z\) and the value \(z = 9\). By substituting \(9\) for \(z\), we convert the expression to \(2 + 4 \times 9\).

The powerful thing about substitution is that it reduces the complexity of an equation.

• Instead of working with abstract symbols, you're working with concrete numbers.
• This method is especially useful when multiple variables are involved, and it paves the way for more straightforward calculations in subsequent steps.

Multiplication

Multiplication is a basic yet vital arithmetic operation that involves taking a number and adding it to itself a certain number of times. In our example, after substituting the value of \(z\) into the expression \(2 + 4 \times 9\), the next step is to perform the multiplication.

Multiplication comes first in the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule is essential to ensure calculations are done correctly.

In our problem, multiplying \(4\) by \(9\) gives us \(36\).

• Think of \(4\) times something as adding \(4\) together \(9\) times, which equals \(36\).
• Mastering multiplication is crucial as it frequently appears in more advanced mathematical problems.

Keep in mind that multiplication is distributive, which means that it can be broken down or expanded, making it very flexible in problem-solving.

Addition

Addition is one of the simplest and most frequently used operations in mathematics. It involves combining numbers to get their total. Once we've handled substitution and multiplication, the expression \(2 + 36\) requires an addition step to finalize the solution.

This step is straightforward: simply add \(2\) to \(36\), resulting in the final answer of \(38\).

• Addition follows multiplication in order of operations, underlining its role in verifying the completion of calculations properly.
• Being fundamental in arithmetic, a solid understanding of addition aids in handling more complex equations effortlessly.

By doing these operations sequentially—substitution, multiplication, and then addition—you can follow a clear path to arrive at the correct answer in math problems.