Problem 45 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ (x-2) \cdot \frac{x-1}{x-2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is `x-1`.

Step 1: Identify common factors

The expression is given as $$ (x-2) \cdot \frac{x-1}{x-2} $$ Here, we can see that `(x-2)` is a common factor in both the numerator and the denominator.

Step 2: Cancel out the common factors

Since `(x-2)` is a common factor in both the numerator and the denominator, we can cancel it out: $$ (x-2) \cdot \frac{x-1}{x-2} = \cancel{(x-2)} \cdot \frac{x-1}{\cancel{x-2}} $$

Step 3: Perform the multiplication

After canceling out the common factor `(x-2)`, we are left with: $$ 1\cdot(x-1) $$ Multiplying this, we get the final simplified expression: $$ x-1 $$

Key Concepts

Canceling Common Factors

In algebra, one crucial skill is the ability to cancel common factors in a fraction or expression. This technique helps simplify expressions and solve problems efficiently. When you have a fraction, check if the same factor appears in both the numerator and the denominator. If it does, that factor can be removed or 'canceled' because it is common to both parts.

Consider the example from the exercise: \( (x-2) \cdot \frac{x-1}{x-2} \). Here, \((x-2)\) is present in both the numerator, as part of \((x-2)\cdot\), and in the denominator \(\frac{...}{x-2}\). As they appear on both top and bottom, they can be canceled out. This simplifies the expression and eliminates unnecessary complexity.

**Steps to Cancel Common Factors:**

• Identify matching terms in both the numerator and the denominator.
• Cross out or remove the factor that appears in both places, treating it as multiplication by 1.
• Recalculate the expression without these terms.

Canceling like this is similar in concept to reducing fractions to their simplest form by dividing by the greatest common factor.

Multiplying Expressions

Multiplying expressions is a fundamental aspect of algebra that involves combining several terms or factors into a single entity. It often eases problem-solving by converting complex expressions into more manageable forms. This process includes distributing terms and applying the basic rules of arithmetic.

**How Multiplication Works in the Example:**

• After canceling the common factor \((x-2)\), the problem simplifies to: \(1\cdot(x-1)\).
• Multiplying it, you follow the rule where anything multiplied by 1 stays the same.
• This highlights why multiplying doesn't alter the expression's core value: \((x-1)\).

Multiplying simple monomials, like in the example, demonstrates the principle of maintaining the expression's identity while using multiplication to tidy up its presentation.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is all about making them easier to understand or work with. A simplified expression is clearer and often more straightforward to use in further calculations.

To simplify an expression, follow these general steps:

• Use any available opportunities to cancel out common factors or terms.
• Perform arithmetic operations, such as multiplication or division, to unify the parts of the expression.
• Rewrite the expression in its simplest form, free of unnecessary components.

In the given example, \((x-2) \cdot \frac{x-1}{x-2}\) starts with a complex appearance. However, upon simplifying by canceling and performing multiplication, you get \(x-1\). This result is now easy to interpret or use in subsequent problems or equations, showcasing the power of simplification to clarify algebraic reasoning.