# Equations

When it comes to understanding algebra, there is no getting around the need to be able to solve simple equations. An equation is a mathematical statement that shows two different measurements, or terms, to be equal. A very basic example of a mathematical equation would be “2 + 2 = 4.” Take note of the “=” sign, as it will be present in every single possible equation, no matter how complex. Equations are most frequently used to solve for one or more missing terms, which are called variables. These are usually represented by letters, such as “x” or “y.”

**Linear Equations**

The most basic type of equations are called linear equations. These equations contain just one simple variable with an exponent of 1. For example: “x + 2 = 4” would be a linear equation.

**Exponential Equations**

Generally speaking exponential equations are any equations in which the variable serves as either the power or the base. For example, “x^{2} + 2 = 4” would be a quadratic equation, as the variable has an exponent of “2,” and “x^{3} + 2 = 4” would be a cubic equation, with an exponent of “3.”

**Third Degree Equations**

These types of equations generally require logarithms to solve unless both sides of the equation contain comparable exponential expressions, which just means that they both have to have either the same base or the same power. For example, “5^{x} = 5^{3}” would be a cubic equation that could be solved without logarithms. Because the bases are the same, the only possible value for the power variable “x” would be “3.”

**Solving Linear Equations**

That’s a bit complicated to start out with, though, so let’s take a look at the simple linear equation: “x + 2 = 4.” In order to solve for “x” in the first example a student would need to isolate the variable, meaning that the “x” would have to appear by itself on one side of the “=” with all remaining elements on the other. This can be done by performing the opposite function of what is currently written to both sides of the equation. In this example that would mean subtracting 2 from both sides, leaving “x = 4 – 2,” or “x = 2” when solved.

**Solving Equation Sets**

An equation set, or system, is a collection of equations that are all dealt with at one time. This means that any given variables should have the same value in each equation. To extend the above example, add to the original equation,“x + 2 = 4,” several more: “x – 1 = 1” and “x + 4 = 6.” Put together they would form an equation set. When it comes to linear equations with just one variable, solving for equation sets is as simple as solving for the variable in one equation and using it throughout the set. However, the most common use for equation sets is in multi-variable systems. For example, “y = 3x – 2” and “y = -x – 6” would be a system of linear equations. In order to solve for y, both equations need to be addressed at the same time. Doing so requires the creation of a graph to determine what numerical value can solve both equations. This number will be represented on the graph as the point where the lines created by both linear equations meet.