Problem 20 - Algebra 2

Question

Simplify each expression. \(\frac{3 t-6}{2-t}\)

Step-by-Step Solution

Answer

The expression simplifies to \(-3\).

Step 1: Identify Common Factors

First, identify any common factors in the numerator \(3t - 6\) and denominator \(2 - t\). In the numerator, both terms have a common factor of 3, which allows us to factor it out: \(3(t - 2)\). The denominator can be rewritten as \( -(t - 2)\) because \(2 - t = -(t - 2)\).

Step 2: Factor the Expression

Express the numerator and the denominator with their factored forms: the numerator as \(3(t - 2)\) and the denominator as \(-(t - 2)\). This reveals a common factor of \((t - 2)\) in both the numerator and denominator.

Step 3: Simplify by Canceling Factors

Cancel the common factor \((t - 2)\) found in both the numerator and denominator, resulting in \( \frac{3}{-1} \), because the factor \((t - 2)\) divides out from the expression.

Step 4: Complete the Simplification

Simplify \( \frac{3}{-1} \) to \(-3\). This is the simplified form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is like cleaning up a messy room. You take an algebraic expression and make it as tidy as possible. The goal is to make the expression easier to work with and understand. For example, you might combine like terms or reduce fractions to their simplest form.

• Look for common terms or patterns that can be grouped together.
• Perform operations within the expression, such as addition or subtraction, to condense it.
• Ensure that fractions are as simple as they can be, which often involves factoring or canceling common terms.

Simplifying makes the expression neat and ready for further mathematical operations or applications.

Factoring

Factoring is a method used to break down an expression into its simplest building blocks. Think of it as taking a big Lego set and finding individual pieces that make it up. In algebra, when you factor an expression, you look for numbers or variables that are shared by all terms in the expression.

• Identify the greatest common factor (GCF), which is the largest number or variable that divides each term.
• Rewrite the expression as a product of the GCF and another term.
• For example, with the expression \(3t - 6\), you can factor out 3 to get \(3(t - 2)\).

After factoring, the expression is easier to simplify or use in further calculations.

Canceling Common Factors

Canceling common factors is similar to reducing fractions. If you have the same factor in both the numerator and the denominator, you can "cross out" or cancel that factor. This can simplify the expression drastically.

• First, ensure both the numerator and denominator are fully factored.
• Look for any factors that appear in both the top and bottom of the fraction.
• Cancel these common factors by dividing them out of the numerator and the denominator.
• For instance, in \(\frac{3(t - 2)}{-(t - 2)}\), you cancel \((t - 2)\), leaving you with \(\frac{3}{-1}\).

This process simplifies the expression, making it less complex and easier to interpret.

Transformation of Expressions

Transformation of expressions involves changing the form of an algebraic expression. This can include factoring, expanding, or other algebraic manipulations to achieve a desired form. The transformation can help solve equations or simplify complex expressions.

• Adjust the expression step by step while maintaining its equivalence.
• Use recognized identities and properties to guide transformations: for instance, converting \(2 - t\) to \(-(t - 2)\).
• This involves strategic alterations to the structure while keeping the value unchanged.
• Ultimately, these transformations allow easier interpretation and application of the expression, resulting in forms like \( -3 \) in our example once the transformation is complete.

The goal is achieving an expression that's easier to work with while retaining its original value or meaning.