Problem 11 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when t = 3. \(4 t^{2}\)

Step-by-Step Solution

Answer

The value of the expression \(4 t^{2}\) when \(t = 3\) is \(36\).

Step 1: Identifying the Variable in the Expression

Look at the given expression, which is \(4 t^{2}\). The variable here is \(t\). The task is to substitute \(t\) with \(3\) and simplify the expression.

Step 2: Substitute Variable

Substitute the variable \(t\) in the expression \(4 t^{2}\) with the value \(3\). So, the expression will become \(4 * 3^{2}\).

Step 3: Evaluate the Expression

The expression \(4 * 3^{2}\) simplifies to \(4 * 9\), which equals \(36\). Hence, the value of the expression is \(36\).

Key Concepts

Understanding Variables

In mathematics, variables are symbols that represent numbers or values that can change. In the expression \(4t^2\), the variable is \(t\).
Variables are placeholders that allow us to write expressions and equations in a general form, which can be used for various calculations.
Using variables makes mathematical expressions flexible. They can represent any number, depending on the situation.
For instance, in our exercise, \(t\) can take any value, but we have been asked to evaluate it for \(t=3\). Recognizing variables in expressions is crucial before performing operations like substitution. This step is the foundation of solving algebraic problems and allow us to manipulate and understand mathematical relationships.

The Process of Substitution

Substitution is the process of replacing a variable with a specific value. In our exercise, we substitute \(t\) with \(3\) in the expression \(4t^2\).
This means wherever we see \(t\), we replace it with \(3\). So, the expression \(4t^2\) changes to \(4 \, \cdot \, 3^2\). Substitution allows us to transform an expression with variables into a numerical expression, making it easier to evaluate.

• Identify the variable to be substituted.
• Locate every instance of the variable in the expression.
• Replace the variable with the given number.

Through substitution, we can shift from conceptual expressions to concrete numbers. This is particularly useful in scenarios requiring precise computation, such as in physics or engineering.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. After substituting the variable in our expression \(4t^2\), we got \(4 \, \cdot \, 3^2\).
The next step is to simplify. This involves performing the arithmetic operations to reach a single value.

• Solve any exponents first. For \(3^2\), calculate \(3 \, \cdot \, 3 = 9\).
• Multiply the result by the remaining coefficient. So, \(4 \, \cdot \, 9 = 36\).

The final simplified expression is \(36\). Simplification is important because it helps us understand and work with expressions more easily. It is a way to condense the problem to its most basic form to get a clear result. This skill is essential in algebra since it allows us to see connections and solutions more clearly.