Problem 78 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{a x+b y+a y+b x}{a+b} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x + y\).

Step 1: Factor the Numerator

Look at the terms in the numerator, which is \( ax + by + ay + bx \). Notice that it can be rearranged and grouped as \( ax + bx + ay + by \). Now, let's factor by grouping: \( x(a + b) + y(a + b) \). This leads us to a common factor.

Step 2: Combine Like Terms

After factoring, you have \( x(a + b) + y(a + b) = (x + y)(a + b) \). This shows the expression within the numerator can be simplified to \((x + y)(a + b)\), revealing a common factor in both the numerator and denominator.

Step 3: Simplify the Expression

The expression is now \(\frac{(x + y)(a + b)}{a + b}\). Since \(a + b\) appears in both the numerator and the denominator, you can cancel \(a + b\) from both to simplify the expression to \(x + y\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, called factors, that when multiplied together produce the original expression. In the context of algebra, factoring helps identify common factors among terms to simplify an expression more easily.
To factor an expression, especially when dealing with a polynomial like the one in our original exercise, you can use the technique of "factoring by grouping."

• Start by rearranging or grouping the terms in the polynomial to make it easier to identify common factors.
• Our expression in the example is rearranged to highlight the factors: \( ax + bx + ay + by \).
• By grouping similar terms, we can factor each pair separately: \( x(a + b) + y(a + b) \).

Here, the common factor \((a + b)\) is factored out, turning the expression into \((x + y)(a + b)\).
Factoring plays a crucial role because it reveals these kinds of common elements and sets the stage for further simplification.

Like Terms

Like terms are terms within an expression that have identical variables raised to the same power. This concept simplifies expressions by combining terms with the same variables. Recognizing like terms is an important step in combining them to form simpler expressions.
Consider the expression \( ax + bx + ay + by \).We can see groups of terms sharing the same variable:

• Terms \( ax \) and \( bx \) are like terms because they have the same variable \( x \).
• Terms \( ay \) and \( by \) are like terms since they both contain the variable \( y \).

To combine them, factor the variable across their coefficients and make them appear as a single term for each pair. As applied in the example,
\( x(a + b) + y(a + b) \) is achieved by combining \((a+b)\) with each like term group.
These combined like terms help us rewrite the expression for simpler computation and reveal opportunities to reduce redundancies.

Canceling Common Factors

Canceling common factors involves simplifying expressions by eliminating identical factors present in both the numerator and the denominator of a fraction.This process vastly simplifies expressions, making them more manageable to work with or analyze.
For the given exercise, the expression becomes:\[\frac{(x + y)(a + b)}{a + b}\].

• We identify that \( (a + b) \) is a factor in both the numerator and the denominator.
• To simplify the expression, \( (a + b) \) can be canceled out.

This step reduces the expression to:\( x + y \),showing a clear and concise result. It is important to remember that we can only cancel factors, not terms, which means this technique is effective only when similar binomial factors are present in both the top and bottom of a fraction.