Problem 55 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The evaluated expression is 45.

Step 1: Understand the Expression

The expression we need to evaluate is \(5y^2\). This means we must square the value of \(y\) and then multiply it by 5.

Step 2: Substitute the Known Value

We are given that \(y = 3\). Substitute 3 for \(y\) in the expression. This gives us \(5(3)^2\).

Step 3: Calculate the Square

Find the square of 3: \(3^2 = 9\).

Step 4: Perform Multiplication

Multiply 5 by the result from Step 3: \(5 \times 9 = 45\).

Key Concepts

Evaluating Expressions

Evaluating an expression in algebra involves substituting the values of the variables into an expression and performing the necessary arithmetic. For example, when given an expression like \(5y^2\) and a set of values for its variables, such as \(x = 1\), \(y = 3\), and \(z = 5\), our goal is to find the numerical value by carefully following each operation indicated by the expression. The steps involved in evaluating expressions are straightforward:

• Identify the variables and the operations to be performed.
• Substitute the values of the variables into the expression.
• Execute arithmetic operations as per the order of operations, typically known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Breaking down the expression and executing each step precisely ensures you reach the correct result. Understanding this process is crucial for success in algebra and beyond.

Substitution Method

The substitution method in algebra is the process of replacing variables in an expression with their given values. This is a fundamental step when you evaluate expressions.Consider the expression \(5y^2\). You have to replace \(y\) with the value 3. This can be seen as turning the variable-based expression into a purely numerical one: from \(5y^2\) to \(5(3)^2\).This method streamlines calculations and removes ambiguity by:

• Clarifying which numbers replace each variable.
• Allowing you to convert complex variable expressions into simpler numerical problems.

Using substitution correctly helps prevent errors and enables you to proceed to direct computation, simplifying the task into smaller, more manageable steps.

Squaring Numbers

Squaring a number means multiplying the number by itself. It's a fundamental concept in mathematics and appears frequently in algebraic expressions. When tasked with \((3)^2\), you're essentially calculating the magnitude of 3 when taken to the power of 2.Let's dive into why and how we square numbers:

• The operation involves repeated multiplication, making it akin to multiplication tables.
• For example, squaring 3: \(3 \times 3 = 9\).

This step, albeit simple, is crucial as any errors here can lead to incorrect results further down the line. Squaring appears widely from basic math to complex equations, so mastering it is essential for building mathematical skills.

Multiplication in Algebra

Multiplication is a core mathematical operation that involves adding a number to itself a certain number of times. In the context of algebra, it frequently applies to constants and variables, helping in both simplifying expressions and evaluating them, as we see in expressions like \(5 \times 9\).Let's break it down:

• Multiplication in algebra can involve numbers, variables, or both. Here, multiplying a constant with a result from previous operations, \(5 \times 9\), results in 45.
• It is crucial to carry out multiplication after handling any exponents first, in keeping with PEMDAS rules, ensuring the correct order of operations.

In essence, understanding multiplication within algebraic contexts unveils deeper insights into more complex mathematical problems, paving the way for more advanced study topics such as polynomials and algebraic fractions.