Problem 60 - Algebra A Combined Function

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5 .\) $$ 2 z^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 50.

Step 1: Substitute the given value of z

We start by substituting the value of \(z = 5\) into the expression \(2z^2\). This simply means replacing \(z\) with 5 in the expression.

Step 2: Calculate z squared

Next, we calculate \(z^2\) by squaring the substituted value of \(z\). So we have: \(5^2 = 25\).

Step 3: Multiply by 2

Finally, multiply the result from the previous step by 2. Thus, we calculate \(2 \times 25 = 50\).

Key Concepts

Substitution Method

The substitution method is a straightforward yet powerful technique used in evaluating expressions. It involves replacing variables in an expression with their given numerical values. This method simplifies calculations and helps reach a solution more efficiently.
For instance, in the exercise provided, we have an expression involving the variable \(z\) and we are given \(z = 5\). So, according to the substitution method, we substitute 5 in place of \(z\) in \(2z^2\). This results in a new expression: \(2 \times 5^2\).
Using substitution:

• Identify the variable and its given value.
• Replace the variable with the value throughout the expression.
• Proceed to solve the newly formed numerical expression.

Substitution simplifies problem solving by converting variable expressions into numbers, making further calculations much easier.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. It means multiplying the base by itself as many times as specified by the exponent.
In our example, once we substitute \(z=5\) in the expression \(2z^2\), we need to deal with the exponentiation \(5^2\). This calculation involves raising the base 5 to the power 2:
- \(5^2\) is calculated as \(5 \times 5 = 25\).

• The base here is \(5\).
• The exponent is \(2\), indicating the base is used twice in multiplication.
• This results in the product 25.

Exponentiation is an essential skill when evaluating expressions, allowing you to simplify repeated multiplication into a concise form.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which operations should be performed in an expression. By following these rules, we ensure that everyone calculates expressions in the same way, getting the same result.
The standard order of operations is dictated by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In our expression \(2z^2\), this means we:

• First, perform the exponentiation \(5^2\).
• Then, multiply the result by 2, which is the last operation.

Thus, the order is to calculate the power first (\(5^2 = 25\)), and then multiply by 2 (\(2 \times 25 = 50\)). Without following the order of operations, we might end up with incorrect results.
Understanding and applying the order of operations ensures consistency and accuracy in mathematical calculations. By breaking down complex expressions into manageable steps, it makes problem solving more approachable.