Problem 34 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{-4 x-4 y}{x+y} $$

Step-by-Step Solution

Answer

The simplified expression is \(-4\).

Step 1: Factor the Numerator

The first step when simplifying a fraction is to look for common factors in the numerator and the denominator. In the numerator, \(-4x - 4y\), we can factor out a \(-4\). This gives us: \(-4(x + y)\).

Step 2: Write the Fraction With the Factored Numerator

Now replace the numerator in our original fraction with the factored form: \(\frac{-4(x+y)}{x+y}\).

Step 3: Cancel Common Factors

Notice that \((x + y)\) is present in both the numerator and the denominator. Therefore, they cancel each other out: \(\frac{-4}{1} = -4\).

Step 4: Write the Final Simplified Expression

After canceling, the final result of simplifying the original expression is \(-4\).

Key Concepts

Factoring

In algebra, factoring is a crucial step in simplifying expressions, particularly when working with fractions. It involves identifying and extracting the greatest common factors from terms in an expression. Think of it as pulling out a common variable or number to make calculations easier.
In the expression \(-4x - 4y\), the key is to look for common elements in both terms. Here, both terms include a \-4\. By factoring out this \-4\, the expression simplifies to \(-4(x + y)\).

• Find common factors in each term of the expression.
• "Factor out" or remove these common factors.
• Rewrite the expression using the factored form.

With the expression now in its factored form, the upcoming steps of canceling out parts of the fraction become much more straightforward.

Common Factors

Identifying common factors is the bridge that leads to factoring and simplifying algebraic expressions. A common factor is any number or variable that divides all terms in an expression evenly.
After factoring the numerator of the fraction \(-4x - 4y\), you determine that \(-4\) is the common factor in both terms, resulting in the expression \(-4(x + y)\).

• Scan the terms quickly for identical variables or numbers.
• See if these elements can be factored out evenly across the terms.
• Apply "factoring out" to simplify expressions step-by-step.

Recognizing common factors streamlines later steps, especially when fractions are involved, making them easier to handle, like preparing it for canceling factors.

Canceling Factors

Canceling factors is an essential skill once the common factors have been identified and expressed. This step essentially involves removing identical factors from the numerator and the denominator in a fraction.
When we express the fraction \(\frac{-4(x+y)}{x+y}\), \(x + y\) appears in both the numerator and the denominator, enabling us to "cancel" them out. This leads to a simplified fraction:

• Identify identical factors in both numerator and denominator.
• Cancel these to simplify the fraction further.
• Result in a more straightforward expression or number.

Once the factors \(x+y\) are eliminated from both the top and bottom, the expression reduces to simply \(-4\). This process highlights the beauty of algebra, where complex expressions can become much simpler through careful manipulation.