Solving Equations

Solving Equations in One Variable

An equation is a mathematical statement formed by placing an equal sign between two numerical or variable expressions, as in 3x + 5 = 11.

A solution to an equation is a number that can be plugged in for the variable to make a true number statement.

Example 1:

Substituting 2 for x in

3x + 5 = 11

gives

3(2) + 5 = 11, which says 6 + 5 = 11; that's true!

So 2 is a solution.

In fact, 2 is the ONLY solution to 3x + 5 = 11.

Some equations might have more than one solution, infinitely many solutions, or no solutions at all.

Example 2:

The equation

x² = x

has two solutions, 0 and 1, since

0² = 0 and 1² = 1. No other number works.

Example 3:

The equation

x + 1 = 1 + x

is true for all real numbers. It has infinitely many solutions.

Example 4:

The equation

x + 1 = x

is never true for any real number. It has no solutions.

The set containing all the solutions of an equation is called the solution set for that equation.

Equation	Solution Set
3x + 5 = 11	{2}
x² = x	{0, 1}
x + 1 = 1 + x	R (the set of all real numbers)
x + 1 = x	(the empty set)

Sometimes, you might be asked to solve an equation over a particular domain. Here the possibilities for the values of x are restricted.

Example 5:

Solve the equation

over the domain {0, 1, 2, 3}.

This is a slightly tricky equation; it's not linear and it's not quadratic, so we don't have a good method to solve it. However, since the domain only contains four numbers, we can just use trial and error.

So the solution set over the given domain is {0, 1}.

Solving Equations in Two Variables

The solutions for an equation in one variable are numbers. On the other hand, the solutions for an equation in two variables are ordered pairs in the form (a, b).

Example:

The equation

x = y + 1

is true when x = 3 and y = 2. So, the ordered pair

(3, 2)

is a solution to the equation.

There are infinitely many other solutions to this equation, for example:

(4, 3), (11, 10), (5.5, 4.5), etc.

The ordered pairs which are the solutions of an equation in two variables can be graphed on the cartesian plane. The result may be a line or an interesting curve, depending on the equation. See also graphing linear equations and graphing quadratic equations.

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