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

$3x+5=11$

$\left\{2\right\}$

${x}^{2}=x$

$\{0,1\}$

$x+1=1+x$

$\text{R}$
(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 $\{0,1,2,3\}$ .
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.
$\begin{array}{l}{0}^{2}=\sqrt{0}=0\\ {1}^{2}=\sqrt{1}=1\\ {2}^{2}\ne \sqrt{2}\\ {3}^{2}\ne \sqrt{3}\end{array}$
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 6:
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 .