Problem 58 - Elementary Algebra

Question

Simplify. $$ 12 x 2 y 2+3 x y-13 x 2 y 2+10 x y $$

Step-by-Step Solution

Answer

The simplified expression is \( -x^2y^2 + 13xy \).

Step 1: Organize Like Terms

First, organize the equation and group the like terms together. This equation consists of two types: \( x^2y^2 \) and \( xy \) terms. Separate them: \( (12x^2y^2 - 13x^2y^2) + (3xy + 10xy) \).

Step 2: Combine Like Terms

Now, combine the like terms. For the \( x^2y^2 \) terms: \( 12x^2y^2 - 13x^2y^2 = -1x^2y^2 \). For the \( xy \) terms: \( 3xy + 10xy = 13xy \).

Step 3: Write the Simplified Expression

Combine the results from Step 2 into one simplified expression. This gives us: \( -1x^2y^2 + 13xy \).

Step 4: Final Simplified Form

Since the \( -1 \) factor in \( -1x^2y^2 \) is redundant, we can simplify the expression to \( -x^2y^2 + 13xy \).

Key Concepts

Combining Like Terms

When we talk about combining like terms in algebra, we mean adding or subtracting terms that have the exact same variables raised to the same powers. This process is essential in simplifying algebraic expressions. Consider this: terms with identical variable parts can be combined just like regular numbers.

• For example, in the expression \(12x + 3x\), both terms are like terms because they share the same variable, \(x\).
• To combine them, simply add the coefficients: \(12 + 3\), resulting in \(15x\).

When you organize your expression, identify terms with similar variables and exponents and group them together. Doing so makes the next step, combining them, straightforward and helps you simplify the overall expression effectively.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial expression to its simplest form by combining like terms.To start, identify terms within the polynomial that can be merged.

• For instance, in \(12x^2y^2 + 3xy - 13x^2y^2 + 10xy\), notice there are terms with \(x^2y^2\) and terms with \(xy\).
• Separate these into groups and simplify each group one at a time.

By combining the like terms, you effectively streamline the expression to its most basic components.The goal is to eliminate unnecessary complexities, highlighting just the fundamental elements of the polynomial.

Like Terms in Algebra

In algebra, like terms are terms that contain the same variables, with corresponding variables raised to the same power.Recognizing and working with like terms is a critical skill, as it enables simplification and helps solve equations more easily.Here's what you need to know:

• Like terms can be summed all together just like you would add numbers.
• For example, \(7xy\) and \(5xy\) are like terms, combining to form \(12xy\).

Being able to spot like terms quickly will make you more efficient in solving algebra problems, as it's the initial step towards simplifying complex expressions or equations.