# 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

$3\left(2\right)+5=11$ , 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

${x}^{2}=x$

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

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.

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

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

## 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 $\left(a,b\right)$ .

Example 6:

The equation

$x=y+1$

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

$\left(3,2\right)$

is a solution to the equation.

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

$\left(4,3\right),\left(11,10\right),\left(5.5,4.5\right),$ 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 .