Problem 5 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$5 \cdot 6 y \text { when } y=5$$

Step-by-Step Solution

Answer

The simplified expression is 150.

Step 1: Identify the Variable

In the given expression \(5 \cdot 6y\), the variable is \(y\). In this case, \(y\) has been given a value of 5.

Step 2: Substitute the Variable

Substitute the value of \(y\) into the expression. So, the expression becomes \(5 \cdot 6 \cdot 5\).

Step 3: Simplify the Expression

Carry out the multiplication operation to obtain the simplified expression. So, \(5 \cdot 6 \cdot 5 = 150\).

Key Concepts

Substitution in Algebraic Expressions

One of the fundamental skills in algebra is learning how to make use of the substitution method. When you are given an algebraic expression that includes variables, substitution means replacing those variables with given numerical values. Let's say you have the expression 5 × 6y, and you're told that y = 5. You then substitute 5 for every instance of y in the expression.

For a clearer illustration, let's break it down:

• Begin by looking at the expression and identifying the variables. In our original problem, the variable is 'y'.
• Next, replace the variable 'y' with its assigned numerical value, which in this case is 5. This turns the expression into 5 × 6 × 5.
• Now you've successfully transformed the algebraic expression into an arithmetic problem!

It is essential to follow the prescribed order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), after substituting to obtain the correct result.

Simplifying Algebraic Expressions

Simplifying an algebraic expression is all about making it as straightforward as possible. This involves performing operations like addition, subtraction, multiplication, and division to combine like terms and eliminate any unnecessary complexity.

To simplify 5 × 6 × 5, which is our example after substitution, you only need to follow one type of operation since it's all multiplication:

• Multiply the numbers in succession or all at once, here, 5 × 6 × 5.
• The result is a single numerical value, 150, which is simpler than the original algebraic expression.

Simplification is a hugely advantageous skill, especially when dealing with more complex expressions that can be reduced significantly, making them easier to work with in subsequent calculations.

Performing Arithmetic Operations

Performing arithmetic operations such as addition, subtraction, multiplication, and division is the cornerstone of evaluating expressions. Once the expression has been set up through substitution and is ready to be simplified, it's time to do the math.

As demonstrated in our example, after replacing 'y' with 5, we multiply the numbers: \(5 \times 6 \times 5\). This process is a fundamental arithmetic operation. Here are some tips to keep in mind:

• Adhere to the order of operations (PEMDAS) for expressions involving multiple types of arithmetic operations.
• Operations must be carried out from left to right — multiplication and division as they appear in the expression, followed by addition and subtraction.
• Double-check your arithmetic calculations to avoid simple mistakes that can lead to incorrect solutions.

Through consistent practice, performing these basic operations becomes instinctive, which is invaluable for more complex problem-solving in algebra.