## Basics

An unknown quantity is called a variable. A variable is usually denoted by a designated letter, such as $$x$$, $$y$$ , $$a$$ , $$b$$ and $$p$$.

Example If you are given a term $$9x^2 + 6x + 3$$,
$$x$$ is the variable
9 is the coefficient of $$x^2$$
6 is the coefficient of $$x$$
3 is the constant.

A constant refers to a number that does not change. It is not a variable.

For the term $$12p^3 + 2q + 7$$, what is the constant?
CORRECT!

WRONG! TRY AGAIN! A number in front of a variable is a coefficient, not a constant.

WRONG! TRY AGAIN! These are variables.

## Simplifying By Like-Terms

First, simplify according to like-terms. See if you can simplify variables, coefficients, or constants. Look for common factors.

Example Simplify $$3z+7z$$.
Solution  $$3z+7z=(3+7)z=10z$$

Example Simplify $$10p-5q$$.
Solution  $$10p-5q=(5)(2p-q)$$

Simplify $$6y + 4y + y$$.
11y

Simplify $$7p + 21q$$.
$$7(p + 3q)$$
Need more practice? Click here to do some math drill on this topic!

When you see fractions, look for common factors in denominators and numerators, and factor out like terms!

Example Simplify $$\dfrac{4y^2+4x+8}{12}$$.

Solution  $$\require{cancel} \dfrac{4y^2+4x+8}{12} \\ \, \\ =\dfrac{4(y^2+x+2)}{12} \\ \, \\ =\dfrac{\cancel{{\color{red}{4}}}(y^2+x+2)}{\cancelto{3}{{\color{red}{12}}}} \\ \, \\ =\dfrac{y^2+x+2}{3}$$

Example Simplify $$\dfrac{7y+14z}{y+2z}$$.

Solution  $$\require{cancel} \dfrac{7y+14z}{y+2z} \\ \, \\ =\dfrac{7(y+2z)}{y+2z} \\ \, \\ =\dfrac{7\cancel{({\color{red}{y+2z}})}}{\cancel{{\color{red}{y+2z}}}} \\ \, \\ =7$$

Simplify $$\dfrac{6x^2+9x+6}{3}$$.

$$\require{cancel} \dfrac{6x^2+9x+6}{3} \\ =\dfrac{3(2x^2+3x+2)}{3} \\ =\dfrac{\cancel{{\color{red}{3}}}(2x^2+3x+2)}{\cancel{{\color{red}{3}}}} \\ =2x^2+3x+2$$
Need more practice? Click here to do some math drill on this topic!

## Using The Properties of 1

Multiplying or dividing by 1 DOES NOT change the equation. Using this property, we can use the same denominator and numerator that is equal to 1 to simplify.

Example Simplify $$\dfrac{6x+3}{\sqrt{3}}$$.

Solution  $$\require{cancel} \dfrac{6x+3}{\sqrt{3}} \\ \, \\ =\dfrac{3(2x+1)}{\sqrt{3}}\cdot\dfrac{\sqrt{3}}{\sqrt{3}} \\ \, \\ =\dfrac{3\sqrt{3}(2x+1)}{3} \\ \, \\ =\dfrac{\cancel{{\color{red}{3}}}\sqrt{3}(2x+1)}{\cancel{{\color{red}{3}}}} \\ \, \\ =\sqrt{3}(2x+1)$$

Simplify $$\dfrac{2x+4}{\sqrt{2}}$$.

$$\require{cancel} \dfrac{2x+4}{\sqrt{2}} \\ \, \\ =\dfrac{2x+4}{\sqrt{2}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}} \\ \, \\ =\dfrac{\sqrt{2}(2x+4)}{2} \\ \, \\ =\dfrac{\sqrt{2}(2)(x+2)}{2} \\ \, \\ =\dfrac{\sqrt{2}(\cancel{{\color{red}{2}}})(x+2)}{\cancel{{\color{red}{2}}}} \\ \, \\ =\sqrt{2}(x+2)$$