Problem 11 - Algebra 2

Question

Evaluate each expression if \(w=6, x=0.4, y=\frac{1}{2},\) and \(z=-3\). \(w(8-y)\)

Step-by-Step Solution

Answer

The expression evaluates to 45.

Step 1: Substitute the Values

First, we need to substitute the given values into the expression. The expression provided is \(w(8 - y)\). We know \(w = 6\) and \(y = \frac{1}{2}\). Substitute these values to get: \(6(8 - \frac{1}{2})\).

Step 2: Simplify Inside the Parentheses

Next, perform the operation inside the parentheses. Calculate \(8 - \frac{1}{2}\), which gives \(8 - 0.5 = 7.5\).

Step 3: Multiply the Constants

Now, we multiply 6 by 7.5 to evaluate the expression. Calculate \(6 \times 7.5 = 45\).

Key Concepts

Substitution Method

The substitution method is a helpful technique used in mathematics to evaluate expressions. It involves replacing variables in an equation or expression with their given values. In our example, we have the expression \(w(8-y)\), where values for \(w\) and \(y\) are provided: \(w = 6\) and \(y = \frac{1}{2}\). By substituting these values, the expression becomes \(6(8 - \frac{1}{2})\). This method simplifies the evaluation process by allowing us to work with numbers instead of variables, making calculations more straightforward.
To apply the substitution method successfully:

• Identify the variables in the expression.
• Note down their corresponding values as provided.
• Replace each variable with its value in the expression.

Substitution is crucial when dealing with expressions involving variables that need concrete numeric values for further evaluation.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform with numbers, including addition, subtraction, multiplication, and division. In the context of our example, arithmetic operations are necessary to evaluate the expression after substitution.First, we focus on the subtraction inside the parentheses in the expression \(6(8 - \frac{1}{2})\). Here, we subtract \(\frac{1}{2}\) from 8, which involves understanding fractions. The result of this operation is 7.5.
After simplifying within the parentheses, we carry out multiplication: multiply 6 by 7.5 to arrive at a final result of 45.
When performing arithmetic operations, keep in mind:

• Respect the order of operations (parentheses first).
• Work step-by-step to avoid mistakes.
• Use exact values especially when dealing with fractions and decimals to maintain precision.

Mastering arithmetic operations is essential for successfully evaluating expressions and solving math problems.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form to make calculations easier. In our exercise, simplifying involves performing the operations inside the parentheses and then multiplying.Initially, we simplify the inner expression of \(8 - \frac{1}{2}\) by converting the fraction to a decimal, yielding 7.5. This simplification helps us to manage calculate operations more efficiently. After obtaining 7.5, the next step is to multiply 6 by 7.5 to fully simplify the expression, leading us to the final result of 45.
Simplifying expressions correctly involves:

• Calculating operations in the correct sequence.
• Reducing expressions inside parentheses as much as possible before moving outside them.
• Converting complex fractions or mixed numbers to simpler forms like decimals when applicable.

By practicing simplifying expressions, you will develop a clearer understanding and efficiency in handling mathematical equations.