Problem 31 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 7 a b-9 a b+4 a b $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(7ab-9ab+4ab\). Answer: \(2ab\)

Step 1: Identify like terms

In the given expression, \(7ab\), \(-9ab\), and \(4ab\) are like terms, as they all contain the same variables \(a\) and \(b\) raised to the same power.

Step 2: Add and subtract like terms

To simplify the expression, we will add or subtract the coefficients of the like terms. In this case: $$ 7ab - 9ab + 4ab = (7 - 9 + 4)ab = 2ab $$ The simplified expression is \(2ab\).

Key Concepts

Like Terms

In algebra, understanding like terms is crucial when dealing with expressions. Like terms are terms that have the exact same variables raised to the same power. This means their variable parts are identical, even if their coefficients (the numbers in front of the variables) are different.

For example, in the expression \(7ab - 9ab + 4ab\), each term has the variables \(a\) and \(b\). Since every term has the same variables and these variables are not raised to different powers, they are considered like terms.

• Like terms simplify expressions by allowing us to combine them using basic arithmetic operations, such as addition or subtraction, to produce a simpler form.
• When identifying like terms, ensure that the coefficients might differ, but the variable part must stay the same.

Once you identify like terms, you can then combine them to simplify the expression.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This makes them easier to work with, especially when solving equations or performing further algebraic manipulation.

To simplify an expression, follow these steps:

• Identify like terms within the expression.
• Combine the coefficients of the like terms by performing addition or subtraction.
• Ensure that any unnecessary terms or elements that cancel each other out are removed.

In the given expression \(7ab - 9ab + 4ab\), each term has been identified as a like term, so we can combine them by performing the arithmetic operation on their coefficients:

• Calculate \(7 - 9 + 4\) to get \(2\).
• Attach the common variable part \(ab\) back: \(2ab\).

The simplified expression becomes \(2ab\). Simplifying in this way reduces clutter and makes solving related algebraic problems more straightforward.

Addition and Subtraction of Terms

Combining terms through addition and subtraction is a fundamental skill in algebra. Whether you're working with numbers or algebraic expressions, the process involves performing operations on the coefficients of like terms while maintaining the variable part unchanged.

In the expression \(7ab - 9ab + 4ab\), the process involves arithmetic operations:

• Add or subtract the coefficients \(7\), \(-9\), and \(4\).
• The steps look like this: \(7 - 9\) gives you \(-2\), then add \(4\) to get \(2\).
• Keep the variable part \(ab\) consistent: \(2ab\).

Once you've completed these calculations, you've effectively combined the terms using addition or subtraction. Remember that the variable(s) do not change - only the coefficients do. This is crucial when dealing with more complex expressions or equations. Consistently applying this principle simplifies solving and understanding algebraic expressions.