Like Terms || Learn how to combine like terms

In this post we will learn about the concept of like terms.
First we will understand the definition, then move on to solve some questions.

To understand the concept you should have basic understanding of constants, variables and coefficients in mathematics.

What are Like Terms?

The entities having same variables raised to the same power are the like terms.

In Like terms, the coefficient (i.e. constants) value can be different.


Given below are rules for like terms:
(a) The entity must have same variables.
(b) The variables should be raised to same power.
(c) The constants ( or coefficients) value can be different.

Examples of Like Terms


(a) 3x and 2x

Both are like terms as:
⟹ both have same variable x
⟹ Both the entities have x with power 1



(b) 5 and 6y

These are not like terms because digit 5 doesn’t have variable y.


(c) \mathtt{5x^{2} y\ \ \&\ \ 19x^{2} y}

Both the entities are like terms as:
⟹ both have same variable x and y
⟹ x is raised to power 2 and y is raised to power 1.


(d) \mathtt{14xy\ \ \&\ \ 14x^{2} y}

These are not like variables.
First entity ⟹ 14 x y
Power of x = 1
Power of y = 1

Second entity ⟹ \mathtt{14x^{2} y}
Power of x = 2
Power of y = 1

Both the entities have different power of x, hence these are not like terms.


(e) \mathtt{2xy^{3} z^{3} \ \ \&\ \ 3xy^{3} z{^{3}}}

First Entity ⟹ \mathtt{2xy^{3} z^{3}}
Power of x = 1
Power of y = 3
power of z = 3

Second entity ⟹ \mathtt{3xy^{3} z{^{3}}}
Power of x = 1
Power of y = 3
power of z = 3

Both the entities have same variables with same power, hence they are like terms.



(f) \mathtt{x^{2} z^{4} \ \ \&\ \ 9x^{2} y{^{4}}}

First entity ⟹ \mathtt{x^{2} z^{4}}
Power of x = 2
Power of z = 4

Second entity ⟹ \mathtt{9x^{2} y{^{4}}}
Power of x = 2
power of y = 4

Both the entities have different variable, hence they are not like terms.

Combining like terms

Like terms can be easily added or subtracted in algebra equation.

While addition or subtraction, only the coefficients of the entities change, rest of the variables remain the same.

In order to combine the terms, do the following steps:

(a) Check if the given terms are like or not

(b) Add or subtract the coefficient as per the questions and leave the variable as it is.

Combining like terms examples

(01) 5x + 6x

Solution
Both the entities are like terms.

Adding the entities, we get;
⟹ 5x + 6x
11x

Note that we have just added the coefficient, the variable x and its power remains the same.

(02) \mathtt{5x^{2} \ +\ 4x\ +\ 3x^{2}}

Solution
In the given expression, only first and third entity are like terms as both have variable x with power 2.

Here we will only add like term entities.

\mathtt{\Longrightarrow \ 5x^{2} \ +\ 4x\ +\ 3x^{2}}\\\ \\ \mathtt{\Longrightarrow \ 5x^{2} +\ 3x^{2} +\ 4x}\\\ \\ \mathtt{\Longrightarrow \ 8x^{2} +4x}

Here we have added the coefficient of like terms, keeping the power of variable as it is.

(03) 7xy + x + y + 9x +xy

Solution
Let’s first combine the like terms in brackets.
⟹ (7xy + xy) + (x + 9x) + y
⟹ 8xy + 10x + y

(04) \mathtt{9x^{2} \ +\ 10x^{3} -\ 6x^{2} +\ 7\ -\ 6x\ +\ 3}

Solution
Let’s combine the like terms and then solve it

\mathtt{\Longrightarrow \ \left( 9x^{2} -\ 6x^{2}\right) +\ 10x^{3} -6x\ +\ ( 7\ +\ 3)}\\\ \\ \mathtt{\Longrightarrow \ 3x^{2} +\ 10x^{3} -6x\ +\ 10}

(05) 10xy + 3yz

Solution

Both the entities are not like terms.
Hence, these terms cannot be added.

Questions on Combining Like Terms


(01) Check if the below entities are like terms or not

(a) 3xy and 2y

(b) \mathtt{13x^{3} z\ \ and\ 15x^{3} z}

(c) \mathtt{\frac{10}{x^{3}} \ \ and\ 10x^{3}}

(d) \mathtt{\frac{16x}{y^{2}} \ \ and\ \frac{2x^{2}}{y^{3}}}

(e) \mathtt{10\frac{z}{x} \ \ \&\ 11\ \frac{z}{x}}

(f) \mathtt{\frac{x}{y} \ \ \&\ \ \frac{y}{x}}


Solution

(a) 3xy and 2y

First entity ⟹ 3xy
power of x =1
power of y = 1

Second entity ⟹ 2y
Power of x = 0
Power of y = 1

Given entities are not like terms since the variable x is not present in the second term.

(b) \mathtt{13x^{3} z\ \ and\ 15x^{3} z}

First entity \mathtt{13x^{3} z}
Power of x = 3
power of z = 1

Second entity \mathtt{15x^{3} z}
Power of x = 3
Power of z= 1

Both the entity have same variable with same power.
hence both the entities are like terms.

(c) \mathtt{\frac{10}{x^{3}} \ \ and\ 10x^{3}}

First entity \mathtt{\frac{10}{x^{3}}}

We can rewrite the entity as:
\mathtt{\frac{10}{x^{3}} \ \Longrightarrow \ 10\ x^{-3}}

Power of x = -3


Second entity \mathtt{\ 10\ x^{3}}

Power of x = 3

Both the entities have different power.
Hence, they are not like terms.

(d) \mathtt{\frac{16x}{y^{2}} \ \ and\ \frac{2x^{2}}{y^{3}}}

First entity \mathtt{\frac{16x}{y^{2}}}

The entity can be rewritten as:
\mathtt{\frac{16x}{y^{2}} \ \Longrightarrow \ 16\ x\ y^{-2}}

Power of x = 1
Power of y = -2


Second entity \mathtt{\frac{2x^{2}}{y^{3}} \ \Longrightarrow \ 2\ x^{2} \ y^{-3}}

Power of x = 2
Power of y = -3

Both the entities have different power of x and y.
Hence they are not like terms.

(e) \mathtt{10\frac{z}{x} \ \ \&\ 11\ \frac{z}{x}}

First entity \mathtt{\frac{10\ z}{x} \ }

The entity can be written as:
\mathtt{\frac{10\ z}{x} \ \Longrightarrow \ 10\ z\ x^{-1}}

Power of z = 1
Power of x = -1


Second entity \mathtt{\frac{11\ z}{x} \ }

Th entity can be rewritten as:
\mathtt{\frac{11\ z}{x} \ \Longrightarrow \ 11\ z\ x^{-1}}

Power of z = 1
Power of x = -1

Both entities have variable with same power.
Hence the given entities are like terms.

(f) \mathtt{\frac{x}{y} \ \ \&\ \ \frac{y}{x}}

First entity \mathtt{\frac{\ x}{y}}

The entities can be rewritten as:
\mathtt{\frac{\ x}{y} \ \Longrightarrow \ x\ y^{-1}}

Power of x = 1
Power of y = -1

Second entity \mathtt{\frac{\ y}{x}}

The entities can be rewritten as:
\mathtt{\frac{\ y}{x} \ \Longrightarrow \ y\ x^{-1}}

Power of x = -1
power of y = 1

The entities have different powers of variables.
Hence they are not like terms

(02) Combine the given terms

(a) 3x + 9x

(b) 9x + 10y + x + 4y

(c) \mathtt{\ 4x^{2} \ +\ 11x -\ 9x^{2} +\ 3y}

(d) \mathtt{x^{2} yz^{2} \ +6xyz +\ 2x^{2} -\ 5x^{2} yz^{2} \ }


Solution

(a) 3x + 9x

Both the given entities are like terms, hence addition is possible.

⟹ 3x + 9x
⟹ 12x

(b) 9x + 10y + x + 4y

Lets combine the like terms in bracket.
⟹ ( 9x + x ) + (10y + 4y)
⟹ 10x + 14y

(c) \mathtt{\ 4x^{2} \ +\ 11x -\ 9x^{2} +\ 3y}

first combine the like term in bracket

\mathtt{\Longrightarrow \ 4x^{2} \ +\ 11x -\ 9x^{2} +\ 3y\ }\\\ \\ \mathtt{\Longrightarrow \left( 4x^{2} \ -\ 9x^{2}\right) \ +\ 11x\ +\ 3y}\\\ \\ \mathtt{\Longrightarrow \ -5x^{2} \ +\ 11x\ +\ 3y}

(d) \mathtt{x^{2} yz^{2} \ +6xyz +\ 2x^{2} -\ 5x^{2} yz^{2} \ }

Let’s combine the like terms in bracket.

\mathtt{\Longrightarrow \left( x^{2} yz^{2} -\ 5x^{2} yz^{2}\right) \ +6xyz +\ 2x^{2}}\\\ \\ \mathtt{\Longrightarrow \ -\ 4x^{2} yz^{2} \ +\ 6xyz\ +\ 2x^{2}}

(e) \mathtt{\frac{3x}{y^{2}} \ \ +\ \frac{2y^{2}}{x}}

Both the terms are not like terms.
hence, they cannot be simplified further.


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