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 .
 CLEP American Government Courses & Classes
 Series 39 Test Prep
 Photometry Tutors
 DAT Test Prep
 FS Exam  Professional Licensed Surveyor Fundamentals of Surveying Exam Courses & Classes
 West Virginia Bar Exam Courses & Classes
 DAT Quantitative Reasoning Tutors
 Series 3 Courses & Classes
 Stratigraphy Tutors
 PRAXIS Courses & Classes
 Exam PA  Predictive Analytics Test Prep
 Math Foundations Tutors
 CLEP Chemistry Tutors
 Catalan Tutors
 Louisiana Bar Exam Test Prep
 Middle School Writing Tutors
 Indiana Bar Exam Test Prep
 Chemistry REGENTS Tutors
 Actuarial Exam STAM Courses & Classes
 GMAT Courses & Classes