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 $3 x + 5 = 11$ .

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

$3 x + 5 = 11$

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 $3 x + 5 = 11$ .

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

Example 2:

The equation

$x^{2} = x$

has two solutions, $0$ and $1$ , since

$0^{2} = 0$ and $1^{2} = 1$ . 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
$3 x + 5 = 11$	${2}$
$x^{2} = x$	${0, 1}$
$x + 1 = 1 + x$	$R$ (the set of all real numbers)
$x + 1 = x$	$\emptyset$ (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

$x^{2} = \sqrt{x}$

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.

$\begin{array}{l} 0^{2} = \sqrt{0} = 0 \\ 1^{2} = \sqrt{1} = 1 \\ 2^{2} \neq \sqrt{2} \\ 3^{2} \neq \sqrt{3} \end{array}$

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