Problem 72 - Elementary Algebra Within Reach

Question

Describe how to combine like terms. Give an example of an expression that can be simplified by combining like terms.

Step-by-Step Solution

Answer

To combine like terms, identify terms that have the same variables with the same powers, then add or subtract the coefficients of these terms. As an example, for the expression \(3x + 7y + 5x - 2y\), the simplified expression after combining like terms would be \(8x + 5y\)

Step 1: Identify Like Terms

The first step is to identify the 'like terms' in the given expression. 'Like terms' are terms that have the same variables with the same powers. For an example, take an expression \(3x + 7y + 5x - 2y\). Here, \(3x\) and \(5x\) are 'like terms', and \(7y\) and \(-2y\) are 'like terms'

Step 2: Combine Like Terms

After identifying the 'like terms', the next step is to combine them. This is simply done by adding or subtracting the coefficients (the numerical part) of the 'like terms'. In our example, combining \(3x\) and \(5x\) gives \(8x\), and combining \(7y\) and \(-2y\) gives \(5y\)

Step 3: Write the Simplified Expression

Once the 'like terms' have been combined, you can write the simplified expression. The simplified expression of our chosen example is \(8x + 5y\)

Key Concepts

Like Terms in Algebra

When dealing with algebraic expressions, it's essential to understand the concept of 'like terms'. Like terms are parts of an expression that have identical variable components. This means that for terms to be considered 'like', they must have the same variable and the variable must have the same power.
For instance, in the expression \(3x + 7y + 5x - 2y\), the terms \(3x\) and \(5x\) are like terms because they both have the variable \(x\) raised to the power of one. Likewise, \(7y\) and \(-2y\) are like terms since they both involve the variable \(y\) also raised to the power of one.

• If two terms are not identical in their variable components, like \(x^2\) and \(x\), they aren't considered like terms even if they have the same base variable.
• Like terms can easily be added or subtracted by combining the coefficients, or the numerical parts, of the terms.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is all about making the expression as concise as possible by combining like terms. This process helps in making the work compatible with further calculations or evaluations.
When you simplify expressions, follow these steps:

• Identify like terms in the expression. This involves finding pairs or groups of terms that have the same variables to the same powers.
• Combine these like terms by adding or subtracting their coefficients. For example, in \(3x + 5x\), you would add the coefficients 3 and 5 to get \(8x\).

After combining like terms, you'll possess the simplest form of the expression like transforming \(3x + 7y + 5x - 2y\) into \(8x + 5y\). A simplified expression is always easier to handle, especially in complex equations.

Identifying Like Terms

The ability to identify like terms quickly and accurately is crucial in algebra. It is the stepping stone to simplifying expressions and solving equations.
To identify like terms, consider the following:

• Look for terms that have exactly the same variable part. This includes checking that the powers of the variables are the same.
• Do not consider the numerical coefficients when identifying like terms; focus solely on matching the variable part.

For instance, in this expression \(3x^2 + 4xy - 2x + x^2 - 5xy\), both \(3x^2\) and \(x^2\) are like terms. Similarly, \(4xy\) and \(-5xy\) are like terms. The variable component \(x^2\) and \(xy\) maintain their identity while focusing purely on the variables and their powers.