1 Addition and subtraction of like terms

Like terms are multiples of the same quantity. For example $5y$ , $17y$ and $\frac{1}{2}y$ are all multiples of $y$ and so are like terms. Similarly, $3x^2$ , $-5x^2$ and $\frac{1}{4}{x}^2$ are all multiples of $x^2$ and so are like terms.

Further examples of like terms are:

$kx$ and $\ell x$ which are both multiples of $x$ ,

$x^{2}y$ , $6x^{2}y$ , $-13x^{2}y$ , $-2y{x}^2$ , which are all multiples of $x^{2}y$

$ab{c}^2$ , $-7ab{c}^2$ , $kab{c}^2$ , are all multiples of $ab{c}^2$

Like terms can be added or subtracted in order to simplify expressions.

Example 27

Simplify $5x-13x+22x$ .

Solution

All three terms are multiples of $x$ and so are like terms. The expression can be simplified to $14x$ .

Example 28

Simplify $5z+2x$ .

Solution

$5z$ and $2x$ are not like terms. They are not multiples of the same quantity. This expression cannot be simplified.

Task!

Simplify $5a+2b-7a-9b$ .

Answer

Example 29

Simplify $2x^2-7x+11x^2+x$ .

Solution

$2x^2$ and $11x^2$ , both being multiples of $x^2$ , can be collected together and added to give $13x^2$ .

Similarly, $-7x$ and $x$ can be added to give $-6x$ .

We get $2x^2-7x+11x^2+x=13x^2-6x$ which cannot be simplified further.

Task!

Simplify $\frac{1}{2}x+\frac{3}{4}x-2y$ .

Answer

Example 30

Simplify $3a^{2}b-7a^{2}b-2b^2+a^2$ .

Solution

Note that $3a^{2}b$ and $7a^{2}b$ are both multiples of $a^{2}b$ and so are like terms. There are no other like terms. Therefore

$3a^{2}b-7a^{2}b-2b^2+a^2=-4a^{2}b-2b^2+a^2$

Exercises

1. Simplify, if possible,
   1. $5x+2x+3x$ ,
   2. $3q-2q+11q$ ,
   3. $7x^2+11x^2$ ,
   4. $-11v^2+2v^2$ ,
   5. $5p+3q$
2. Simplify, if possible,
   1. $5w+3r-2w+r$ ,
   2. $5w^2+w+1$ ,
   3. $6w^2+w^2-3w^2$
3. Simplify, if possible,
   1. $7x+2+3x+8x-11$ ,
   2. $2x^2-3x+6x-2$ ,
   3. $-5x^2-3x^2+11x+11$ ,
   4. $4q^2-4r^2+11r+6q$ ,
   5. $a^2+ba+ab+b^2$ ,
   6. $3x^2+4x+6x+8$ ,
   7. $s^3+3s^2+2s^2+6s+4s+12$ .
4. Explain the distinction, if any, between each of the following expressions, and simplify if possible.
   1. $18x-9x$ ,
   2. $18x\left(9x\right)$ ,
   3. $18x\left(-9x\right)$ ,
   4. $-18x-9x$ ,
   5. $-18x\left(9x\right)$
5. Explain the distinction, if any, between each of the following expressions, and simplify if possible.
   1. $4x-2x$ ,
   2. $4x\left(-2x\right)$ ,
   3. $4x\left(2x\right)$ ,
   4. $-4x\left(2x\right)$ ,
   5. $-4x-2x$ ,
   6. $\left(4x\right)\left(2x\right)$
6. Simplify, if possible,
   1. $\frac{2}{3}{x}^2+\frac{x^2}{3}$ ,
   2. $0.5x^2+\frac{3}{4}{x}^2-\frac{11}{2}x$ ,
   3. $3x^3-11x+3yx+11$ ,
   4. $-4\alpha x^2+\beta x^2$ where $\alpha$ and $\beta$ are constants.

Answer