Problem 63 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \begin{equation} (t)(5) \text { when } t=15 \end{equation}

Step-by-Step Solution

Answer

The value of the expression \((t)(5)\) when \(t = 15\) is 75.

Step 1: Understand the Expression

The given expression is \((t)(5)\), which is equivalent to \(5t\). Here, \(t\) is a variable.

Step 2: Substitute the given value

The problem states that \(t = 15\). So, replace \(t\) with 15 in the expression, giving us \((15)(5)\).

Step 3: Solve the Expression

Upon substituting, we now simply have a multiplication operation to perform. So, \((15)(5)\) is equal to 75.

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept in algebra where you replace a variable in an expression with a specific value. Think of a variable as a placeholder or a symbol that can represent any number. This is crucial, as variables allow you to create general expressions that apply to multiple scenarios.
In our exercise, the variable is \( t \), and it's given that \( t = 15 \). Here's what substitution looks like:

• Identify the variable in the expression: In this case, \( t \).
• Replace \( t \) with 15: The expression \((t)(5)\) becomes \((15)(5)\).

This process is called substitution because you're systematically substituting the value of \( t \) in place of the symbol \( t \). By doing this, you simplify the expression to a form that can be evaluated. This transformation is key to solving expressions when specific values are provided.

Multiplication

Once you've substituted the variable with its value, the next step is performing multiplication, a basic arithmetic operation. Multiplying numbers can be thought of as adding a number to itself a certain number of times.
In the context of this exercise, after substituting \( t \) with 15, the expression becomes \((15)(5)\). Here are some quick tips for multiplication:

• Identify which numbers you are multiplying: Here, it's 15 and 5.
• Think of 15 times 5 as adding 15 to itself 5 times (i.e., 15 + 15 + 15 + 15 + 15).

The result is 75.
Multiplication is essential in evaluating expressions because it allows you to handle terms that involve more than simple addition or subtraction.

Evaluating Expressions

Evaluating expressions combines all the steps you've learned—substitution and multiplication—in order to find the final numerical value of an algebraic expression. To "evaluate" means to calculate the expression as fully as possible.
Here's a simple breakdown of how to evaluate the expression \((t)(5)\) when \(t = 15\):

• Substitute the given value into the expression: \((15)(5)\).
• Perform the multiplication: Multiply 15 by 5 to get 75.

The resulting number, 75, is what we call the "value" of the expression when evaluated at \(t = 15\). By following the steps of substitution and multiplication, evaluating algebraic expressions becomes a straightforward process that reveals the expression's numerical value.