Problem 46 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. (Lesson 1.3) $$3 \cdot 15 y \text { when } y=2$$

Step-by-Step Solution

Answer

The evaluated result of the expression \(3 \cdot 15y\) when \(y=2\) is 90.

Step 1: Substitute value for variable

Substitute the given value of \(y\), which is 2, into the expression \(3 \cdot 15y\). This gives us \(3 \cdot 15 \cdot 2\).

Step 2: Simplify multiplication

Simplify the multiplication expression. Here, perform the multiplication in order as \(3 \cdot 15\) first and then multiply the result with 2. Therefore, it can be presented as \((3 \cdot 15) \cdot 2 = 90\).

Key Concepts

Substituting Variables

Understanding how to evaluate expressions requires mastery in substituting variables, a fundamental aspect of algebra. When we have an expression like 3 × 15y, and we are given a specific value for y, in this case, y=2, we substitute, or replace, the variable y with its given value. This substitution process turns our algebraic expression into a straightforward arithmetic problem.

To accurately substitute, first identify the variable and then replace it everywhere it appears in the expression with the provided value, ensuring proper alignment of numbers for the multiplication that follows. It's like a recipe where an ingredient (the variable) changes, and we want to see how it affects the overall dish (the expression).

In the given exercise, the value 2 is substituted for y, converting the expression to 3 × 15 × 2. This direct replacement sets the stage for simplification through multiplication.

Simplifying Multiplication

Once variables have been substituted and the algebraic expression is converted into a numerical one, the next key step is simplifying multiplication. Simplifying isn't just about finding the answer; it’s about understanding the process.

Order of Operations

Multiplication, like other arithmetic operations, follows a specific order. In most cases, you can multiply the numbers in any sequence, but sometimes, particularly in more complex expressions, it's important to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Multiplication Strategies

There are different strategies to simplify multiplication, such as:

• Performing operations step-by-step to avoid errors.
• Breaking down larger numbers into smaller factors for easier calculation.
• Grouping numbers in pairs that are easy to multiply.

For our example, multiply 3 × 15 first to get 45, and then multiplying that result by 2 yields 90. This systematic approach helps avoid mistakes and makes complex multiplications manageable.

Algebraic Expressions

The beauty of algebra lies in its use of algebraic expressions, which are combinations of variables, numbers, and operations. They represent values that can change or vary, which is why understanding algebraic expressions is crucial to solving algebra problems.

An algebraic expression, such as 3 × 15y, tells us that there is a relationship between numbers and variables that can be manipulated through mathematical operations. The expression itself contains several components:

• Coefficients: The numerical factors of variables (in this case, 3 and 15).
• Variables: Symbols that represent unknown or changing values (here, y).
• Constants: Numbers that do not change (for this expression, there are no constants apart from the coefficients).

Comprehending these components is key to evaluating and simplifying expressions. In algebra, the ultimate goal is to manipulate these expressions using mathematical operations to find the values of variables, just as we did by substituting y = 2 to find that the expression 3 × 15y evaluates to 90 when y is replaced with 2.