Simplifying algebraic expressions by combining like terms

Algebra is the branch of mathematics that deals with mathematical expressions. It comprises variables, coefficients, operators, and constants. Example: Let 3x + 4 be an algebraic expression. Then 3 is the coefficient, x is the variable, + is the operator, and 4 is the constant. Algebra helps to derive unknown quantities. Let us go through the components of algebra one by one. 

• Variable: is the quantity that is unknown. It is not fixed. We evaluate the value of the variables based on some other conditions. 
• Coefficient: is the quantity that is multiplied by the variable.
• Operator: operators are +, -, / ×.
• Constant: is the fixed quantity and does not have any variables associated with it. 

Terms

A term is a number, variable product of two numbers, product of two variables, or product of a variable and a coefficient. For example: In the expression x – 4y, x and -4y are terms. Terms can be categorized into two parts,

• Like Terms: The terms that have the same power and same variables. The coefficients can vary but the condition is variables and their exponents should be the same. We basically combine the like terms and help in the simplification of algebraic expressions. For example: 4x + 6x = 10x. Here the variables are some, coefficients are different and the powers are also the same.
• Unlike Terms: The terms that have different powers, different variables, and coefficients can also vary. They cannot be simplified. For example: 4x + 6y + 2x2 the variables, power are different.

How to Combine Like Terms and simplify?

Answer:

The terms that have the same variable and same powers can be simplified. We first rearrange the whole expression by combining like terms and unlike terms. Like terms are segregated on one side and unlike terms are kept on another side. Then the operation of like terms is performed. For example: Evaluate x2 + 3x + y2 +4x. In this expression, the power of x is the same so we perform an addition operation of 4x and 3x  which is 7x. The resultant expression is x2 + 7x + y2. 

Sample Questions

Question 1: Evaluate 3x + 4x + 5x2 + 7x2 + 4x.

Solution:

Combine the like terms of x and x2. 

3x + 4x + 4x = 11x

5x2 + 7x2 = 12x2

The result is 11x + 12x2 

Question 2: Solve 4a + 3b +5c +6a + 9d.

Solution:

Combine the like terms,

4a + 6a = 10a

The result is 10a + 3b +5c + 9d.

Question 3: perform subtraction of x3 – y from x2 – 2y -9y.

Solution: 

x2 – 2y – 9y – (x3 – y) 

= x2 -11y – x3 +y 

= x2 -10y – x3

Question 4: Find the sum of like terms 5ab + 6ab + 7ab + 8ba + 90ba. Hence find the value when a = 1, b = 2.

Solution:

As we all know multiplication is commutative so ba = ab

Therefore in the given question, all terms are like terms. The addition of these terms is 116ba

Therefore putting a = 1, b = 2

The answer is 116 × 1 × 2 = 232.

Question 5: Solve for x if the angles of a triangle are 8x, 3x, and 4x.

Solution:

All the angles are like terms

Sum of angles of triangle is 180°

8x + 3x + 4x = 180

=> 15x = 180

=> x = 12

The angles are 96, 36, 48.

Question 6: If f(x) = x2 + 5x + 9 and g(x) = 3x2 + 9y + 9x find f(x) + g(x).

Solution:

The like terms in f(x) and g(x) are x2 and x so we perform the addition,

x2 + 3x2 + 5x + 9x + 9y + 9 

= 4x2 + 14x + 9y + 9

Question 7: Find the condition test for equality if f(x) = (a – b)2 and g(x) = (a + b)2

Solution:

f(x) = g(x)

=> (a – b)2 = (a + b)2

=> a2 + b2 -2ab = a2 + b2 + 2ab

Performing operations of the like terms we get,

=> 4ab = 0

This is the condition for the equality test of two functions. 

The Order of Operations and Variables:

Even without knowing what a variable is, we can sometimes make expressions with variables look simpler. This is done by simplifying our expression.

Here is a vocabulary word that will help you understand the lesson better:

• Coefficient = the number being multiplied to a variable (in 2n, 2 is the coefficient)
• Reduce = combine or simplify by doing whatever operations we can
• Term = a part of an expression separated from the rest by addition (in 3a + 6b, 3a is one term and 6b is another term)
• Like Terms = any terms in an expression where the variables are the same (3a and 4a, \(2{\text{b}}^{2}\) and \(5{\text{b}}^{2}\), note that \(2{\text{b}}^{2}\) and 3b are not like terms)

Video Source (09:10 mins) | Transcript

Remember to follow the order of operations. Sometimes this means to use the distributive property to solve what’s in the parentheses.

When we see two different letters, we can easily know that we don’t have like terms, but can we add \(3{\text{a}} + 4{\text{a}}^{2}\) ? Let’s say \({\text{a}}=3\), then \({\text{a}}^{2}=9\). Because these are different numbers the answer is no, we cannot add \(3{\text{a}}+4{\text{a}}^{2}\). Any time we have different letters as our variables, or the same letter with different powers, we do not have like terms.

Additional Resources

• Khan Academy: Intro to Combining Like Terms (04:32 mins, Transcript)
• Khan Academy: Simplifying Expressions (04:06 mins, Transcript)
• Khan Academy: Combining Like Terms - Challenge Problem (04:38 mins, Transcript)

Practice Problems

Simplify the following expressions:

1. 7w − 2w


2. 5s − 7 − 3s + 11


3. 5a − 2b − 6 + 3a + 6b


4. \(2{\text{v}}^{2}+6+3{\text{v}}{-}3{\text{v}}^{2}\)


5. \( 2(3-2{\text{t}}) + 5 ({\text{t}} + 3) \)


6. \( ( 4 {\text{x}} + 3 {\text{y}} - 2{\text{z}} ) - 2 ( {\text{x}} + 3 {\text{z}}) \)

What does it mean to combine like terms in an algebraic expression?

What will be the simplified form of 21b − 32 7b − 20b?