Equations state that two expressions have the same value when the equation's variables have specific values, called the solution.
To find the solution to an equation with one variable, apply successive simplification steps until the equation is reduced to a statement of the solution, variable = value.
Multiplying each side of the equation by the same value produces an equivalent equation with the same solution. Adding the same value to each side also produces an equivalent equation. This can be used to manipulate the equation to find its solution.
The order of an equation determines how many solutions it has. The order is the highest exponent of the variable. Generally, the number of solutions equals the order.
Equations of order two or above can be solved by factoring, rearranging the equation to fit the standard form of a set of factors which when multiplied together, equal zero. The equation is satisfied when any of the factors is zero; therefore the solutions to the equation are found by setting each factor to zero in turn and solving. Second order equations can also be solved using quadratic formula.
Equations with more than one unknown variable do not have a single solution. To find a single solution, additional constraints in the form of other equations that must be satisfied with the same solution are required. Generally, a unique equation is needed for each variable. For example, three equations are needed to find a solution involving three variables.
Michael A. Gipe
Wharton Boot Camp