Problem 47 - Elementary Algebra

Question

Simplify. $$ 4 x y-6+2 x y+8 $$

Step-by-Step Solution

Answer

The expression simplifies to \(6xy + 2\).

Step 1: Separate Like and Unlike Terms

Identify and group the like terms and unlike terms in the expression. The like terms here are the terms with the variable "\(xy\)", i.e., \(4xy\) and \(2xy\). The constants are \(-6\) and \(8\).

Step 2: Simplify the Like Terms

Add the coefficients of the like terms. The like terms are \(4xy\) and \(2xy\). Therefore, \(4 + 2 = 6\). Thus, the terms combine to \(6xy\).

Step 3: Simplify the Constants

Add the constant terms \(-6\) and \(8\). Therefore, \(-6 + 8 = 2\).

Step 4: Combine the Simplified Terms

Combine the results from Step 2 and Step 3 to get the final simplified expression. Thus, the expression becomes \(6xy + 2\).

Key Concepts

Like Terms

In algebra, like terms are terms that have the exact same variables raised to the same power. These terms can be combined through addition or subtraction. In the expression \(4xy - 6 + 2xy + 8\), the like terms are those involving "xy."

• Like terms: \(4xy\) and \(2xy\).
• Both terms contain the variable part "xy" making them like terms.

To simplify, combine these like terms by adding or subtracting their coefficients.

Coefficients

The coefficient is the numerical factor of a term that contains a variable. In the expression \(4xy\), the number 4 is the coefficient. Similarly, in \(2xy\), the number 2 is the coefficient.When combining like terms, like in this exercise, add their coefficients:

• Coefficients of \(4xy\) and \(2xy\) are 4 and 2 respectively.
• Add these together to simplify: \(4 + 2 = 6\).

This results in the term \(6xy\), showing how coefficients affect the simplification process.

Constants

Constants are terms in an expression or equation that do not contain any variables; they are simply numbers. In our given expression, the constants are \(-6\) and \(8\).

• These numbers don't change because they do not include a variable.
• To simplify the expression, add these constant terms together: \(-6 + 8 = 2\).

Understanding constants helps in distinguishing which parts of an expression can be directly combined or changed.

Expression Simplification

Expression simplification involves combining like terms and constants to make the expression easier to work with. In this exercise, after identifying and combining the like terms and constants, the expression becomes \(6xy + 2\).Consider why simplification is useful:

• It makes complex expressions clearer and simpler.
• Simplified expressions are often necessary for solving equations or further algebraic work.

By breaking the process into steps, like identifying coefficients and constants, simplification becomes straightforward and manageable.