Problem 118 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ t-t-t-t $$

Step-by-Step Solution

Answer

The simplified expression is \(-4t\).

Step 1: Understand the Expression

The given expression is \( t - t - t - t \). It involves subtracting the same variable multiple times.

Step 2: Simplify the Expression

Subtract \( t \) four times. Start with zero and subtract \( t \) four times: \( 0 - t - t - t - t \). This can be mathematically represented as \( 0 - 4t \) or simply \( -4t \).

Step 3: Verify the Simplification

Check the math by considering the expression \( t + (-t) + (-t) + (-t) + (-t) \), which adds to \( 0 - 4t = -4t \). Thus, the simplification is verified.

Key Concepts

Variables in Algebra

Algebraic expressions often include variables, which are symbols representing numbers whose exact values might not be initially known. In this exercise, the variable is \( t \). Understanding the role of variables is crucial because they allow us to formulate general mathematical rules and solve problems with unknown values.

Variables behave like placeholders and can represent any number. In algebra, letters such as \( x, y, \) or \( t \) stand for these variables. They enable us to operate universally on numbers without specifying which number immediately. This flexibility makes algebra an essential tool for solving equations and modeling real-world scenarios.

When you see a variable repeated in an expression, like in \( t - t - t - t \), it indicates that the same quantity is being used multiple times. To simplify the expression, recognizing the uniform variable helps combine terms efficiently.

Subtraction in Algebra

Subtraction in algebra works similarly to subtraction in arithmetic. It involves taking one value away from another. However, when dealing with algebraic expressions, these values can be variables.

Consider the expression \( t - t - t - t \). Here, we are subtracting the variable \( t \) from itself repeatedly. This results in reducing the total value by increments of \( t \).

In an algebraic context, subtraction helps simplify expressions by combining like terms. For example, knowing that \( t - t \) results in zero is fundamental when simplifying. In our exercise, subtracting \( t \) four times can be seen as the expression "0 minus four times \( t \)," or \( -4t \). This simplification shows how subtraction can transform complex expressions into simpler terms.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division are the building blocks of algebra. These operations can be applied to numbers as well as algebraic variables.

In our example, subtraction is the primary operation used. However, arithmetic operations in algebra aren’t isolated. They often combine to help reduce expressions to their simplest forms. For \( t - t - t - t \), this involves recognizing the repeated subtraction as symbolic of multiplying \( t \) by -1 four times, leading to \( -4t \).

Operations follow a logical order. In algebra, consistently applying this order ensures accurate simplifications:

• Addition and Subtraction: Identify like terms and reduce or simplify sums and differences.
• Multiplication: Used for grouping repeated additions. For instance, subtracting \( t \) four times is equivalent to multiplying \( -1 \) by \( 4t \) to get \( -4t \).
• Division: Used to break down expressions into smaller, manageable parts, especially relevant when simplifying more complex algebraic fractions.

Understanding and applying these operations correctly is key to solving algebraic equations and expressions effectively.