Solving equations is the bread and butter of mathematics. Adding, subtracting, multiplying and dividing numbers are necessary elements of computation, but the real magic lies in being able to find an unknown number given sufficient numerical information to carry this out.
Equations contain variables, which are letters or other non-numerical symbols representing values it is up to you to determine. The complexity and depth of understanding required to solve equations ranges from basic arithmetic to higher-level calculus, but finding the missing number is the goal every time.
The One-Variable Equation
In these problems, you are looking for a unique solution to a problem. For example:
2x + 8 = 38
The first step in these simple equations is isolating the variable on one side of the equal sign, by adding or subtracting a constant as needed. In this case, subtract 8 from both sides to get:
2x = 30
The next step is to get the variable by itself by stripping it of coefficients, which requires division or multiplication. Here, divide each side by 2 to get:
x = 15
The Simple Two-Variable Equation
In these equations, you are actually looking not for a single number but a set of numbers, that is, a range of x-values that correspond to a range of y-values to yield a solution that is a curve or a line on a graph not a single point. For example, given:
y = 6x + 9
You can start by plugging in x-values of your choice. It is convenient to start with 0 and work up and then down by units of 1. This gives
y = 6(0) + 9 = 9
y = 6(1) + 9 = 15
y = 6(2) + 9 = 21
And so on. You can then plot the graph of this equation, or function, if you wish.
The Complicated Two-Variable Equation
This type of problem is a variant on the above, with the wrinkle that neither x not y is presented in simple form. For example, given:
3y - 6 = 6x + 12
You have to choose a plan of attack that isolates one of the variables by itself, free of coefficients.
To start, add 6 to each side to get:
3y = 6x + 18
You can now divide each term by 3 to get y by itself:
y = 2x + 6
This leaves you at the same point as in the previous example, and you can work forward from there.