In this section, the following topics are discussed with examples

The main concept that distinguishes algebra from arithmetic is that of a variable, which is a letter that represents a quantity whose value is unknown. The letters x and y are often used as variables, although any letter can be used. Variables enable you to present a word problem in terms of unknown quantities by using algebraic expressions, equations, inequalities, and functions. This section reviews these algebraic tools and then progresses to several examples of applying them to solve real-life word problems. The section ends with coordinate geometry as other important algebraic tools for solving problems

In the expression w^{3}z + 5z^{2} – z^{2} + 6, the terms 5z^{2} and –z^{2} are called like terms because they have the same variables, and the corresponding variables have the same exponents. A term that has no variable is called a constant term. A number that is multiplied by variables is called the coefficient of a term.

For example, in the expression 2x^{2} + 7x – 5, 2 is the coefficient of the term 2x^{2}, 7 is the coefficient of the term 7x, and -5 is a constant term. A statement of equality between two algebraic expressions that is true for only certain values of the variables involved is called an equation. The values are called the solutions of the equation. The following are examples of some basic types of equations.

**3x ****+ 5 = -2** A linear equation in one variable, x

**x – 3y ****= 10** A linear equation in two variables, x and y

**20y ^{2} +6y**

**-17=0**A quadratic equation in one variable,y

A linear equation is an equation involving one or more variables in which each term in the equation is either a constant term or a variable multiplied by a coefficient. None of the variables are multiplied together or raised to a power greater than 1. For example, 2x + 1 = 7x and 10x – 9y – z = 3 are linear equations, but x+y2 =0 and xz=3 are not.

## Men are liars. We'll lie about lying if we have to. I'm an algebra liar. I figure two good lies make a positive.

TIM ALLEN

To solve an equation means to find the values of the variables that make the equation true, that is, the values that satisfy the equation. Two equations that have the same solutions are called equivalent equations. For example, x + 1 = 2 and 2x + 2 = 4 are equivalent equations; both are true when x = 1 and are false otherwise. The general method for solving an equation is to find successively simpler equivalent equations so that the simplest equivalent equation makes the solutions obvious.

The following rules are important for producing equivalent equations.

• When the same constant is added to or subtracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original equation.

• When both sides of an equation are multiplied or divided by the same nonzero constant, the equality is preserved and the new equation is equivalent to the original equation.

#### Example

**Solve 3x – 4 = 2(x + 4) – 2x**

**Sol:**

3x – 4 = 2x + 8 – 2x

3x – 4 = 8

3x = 8+4

3x = 12

3x = 12 / 3 .’. x=4 .