Coefficient: Collecting Like Terms
The Building Blocks: Terms, Coefficients, and Constants
Before we can collect like terms, we need to understand the basic vocabulary. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like $x$ or $y$), and operators (like add, subtract).
- Term: A single number, variable, or numbers and variables multiplied together. In the expression $5x + 2y - 7$, the terms are $5x$, $2y$, and $-7$.
- Coefficient: The numerical factor of a term. It's the number that is multiplying the variable. In the term $5x$, the coefficient is $5$. If a term looks like just a variable, such as $x$, its coefficient is $1$.
- Constant: A term with no variable part. It is a fixed number. In $5x + 2y - 7$, the constant is $-7$.
What Are Like Terms?
Like terms are terms whose variables (and their exponents) are identical. The coefficients can be different. Think of it like fruits: you can add apples to apples, but not apples to oranges.
| Like Terms | Why? | Unlike Terms | Why? |
|---|---|---|---|
| $5x$ and $-2x$ | Same variable $x$ | $5x$ and $5y$ | Different variables |
| $3x^2$ and $7x^2$ | Same variable and exponent | $3x^2$ and $3x$ | Different exponents |
| $-8$ and $15$ | Both are constants | $4xy$ and $4x$ | Different variable combinations |
The Step-by-Step Process of Collecting Like Terms
Collecting like terms, also known as combining like terms, is the process of simplifying an algebraic expression by adding or subtracting the coefficients of terms that are alike. Follow these steps:
- Identify: Scan the expression and identify all the different groups of like terms. It often helps to assign a different color or symbol to each group.
- Group: Mentally or physically rewrite the expression so that the like terms are next to each other. Remember to move the sign ($+$ or $-$) that is in front of each term.
- Combine: Add or subtract the coefficients of the grouped like terms.
- Write the Simplified Expression: Write the new coefficient with the common variable part. Do not add or change the exponents.
Practical Application: From Simple to Complex Expressions
Let's see how this process works with expressions of increasing difficulty. This is where we put theory into practice.
Example 1: Basic Combination
Simplify $7x + 2y - 3x + 5$.
- Identify: The like terms are $7x$ and $-3x$. The term $2y$ has no other like term, and $5$ is a constant.
- Group: $(7x - 3x) + 2y + 5$
- Combine: $(7 - 3)x = 4x$
- Simplified Expression: $4x + 2y + 5$
Example 2: Dealing with Negative Coefficients and Constants
Simplify $4m - 2n + 5 - m + 3n - 8$.
- Identify: Like terms for $m$: $4m$ and $-m$. For $n$: $-2n$ and $3n$. Constants: $5$ and $-8$.
- Group: $(4m - m) + (-2n + 3n) + (5 - 8)$
- Combine: $(4 - 1)m = 3m$, $(-2 + 3)n = 1n$ (or just $n$), $(5 - 8) = -3$.
- Simplified Expression: $3m + n - 3$
Example 3: Terms with Exponents
Simplify $2x^2 + 5x + 3x^2 - x + 4$.
- Identify: Like terms for $x^2$: $2x^2$ and $3x^2$. For $x$: $5x$ and $-x$. The constant is $4$.
- Group: $(2x^2 + 3x^2) + (5x - x) + 4$
- Combine: $(2 + 3)x^2 = 5x^2$, $(5 - 1)x = 4x$.
- Simplified Expression: $5x^2 + 4x + 4$
Example 4: A More Complex Scenario
Simplify $3a + 4b - 2ab + 5a - 7b + ab$.
- Identify: Like terms for $a$: $3a$ and $5a$. For $b$: $4b$ and $-7b$. For $ab$: $-2ab$ and $ab$.
- Group: $(3a + 5a) + (4b - 7b) + (-2ab + ab)$
- Combine: $8a$, $-3b$, $(-2 + 1)ab = -1ab$ (or $-ab$).
- Simplified Expression: $8a - 3b - ab$
Common Mistakes and Important Questions
Q: Can I combine $x^2$ and $x$?
Q: What is the coefficient of a term that doesn't have a number, like $y$ or $-z$?
Q: Why is it so important to pay attention to the sign (plus or minus) in front of a term?
Footnote
[1] Distributive Property: A fundamental property of algebra stated as $a(b + c) = ab + ac$. When collecting like terms, we use it in reverse: $ab + ac = a(b + c)$.