Graphs of Equations as Graphs of Solutions
A solution to an equation can be plotted on graphs to better visualize how the equation, or function, behaves.
Learning Objective

Recognize that graphing an equation involves graphing solutions to it
Key Points
 To solve an equation is to find what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an equation.
 Once an equation has been graphed, solutions to any particular
$x$ or$y$ value can be readily found by simply looking at the graph.  To solve for a variable of an equation, you must use algebraic manipulations to get the variable by itself on one side of the equation (typically the left).
Terms

expression
An arrangement of symbols denoting values, operations performed on them, and grouping symbols (e.g.,
$(2x+4)$ ). 
equation
An assertion that two expressions are equivalent (e.g.,
$x=5$ ). 
graph
A diagram displaying data, generally representing the relationship between two or more quantities.
Full Text
In mathematics, to solve an equation is to find what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an equation (two expressions related by equality). Each of the expressions contain one or more unknowns.
What is the graphical difference between equations with one variable and equations with two variables?
Graphs of Linear Equations with One Variable
A linear equation in one variable can be written in the form
For example, to graph the equation

$(1,3)$ 
$(1,2)$ 
$(1,1)$ 
$(1,0)$ 
$(1,1)$ 
$(1,2)$ 
$(1,3)$
These can also be found by solving the equation of the graph for
The same is true for an equation written as
Therefore, when the equation is
Graphs of Equations with Two Variables
The graph of a cubic polynomial has an equation like
Plot specific points by substituting chosen
Let's choose values for
Therefore,
After substituting the rest of the values, the following ordered pairs are found:

$(1,8)$ 
$(0,0)$ 
$(1,8)$ 
$(2,10)$
After graphing the ordered pairs and connecting the points, we see that the set of (infinite) points follows this pattern:
Graph of $y=x^39x$
Since the exponent of
Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Graphing Quadratic Equations in Vertex Form, and Direct Variation
 difference
 Appears in these related concepts: The Order of Operations, Factoring a Difference of Squares, and Basic Operations
 function
 Appears in these related concepts: Functions and Their Notation, What is a Linear Function?, and The Vertical Line Test
 linear
 Appears in these related concepts: Factoring General Quadratics, Exponential Growth and Decay, and Graphs of Linear Inequalities
 linear equation
 Appears in these related concepts: Rates of Change, Solving Systems of Equations Using Matrix Inverses, and Linear Inequalities
 ordered pair
 Appears in these related concepts: Applications of Systems of Equations, The Substitution Method, and The Cartesian System
 point
 Appears in these related concepts: Graphing Quadratic Equations In Standard Form, Graphing Equations, and Polynomial and Rational Functions as Models
 polynomial
 Appears in these related concepts: Domains of Rational and Radical Functions, The Remainder Theorem and Synthetic Division, and Finding Polynomials with Given Zeros
 real number
 Appears in these related concepts: Zeros of Polynomial Functions with Real Coefficients, Solving Problems with Inequalities, and Introduction to Complex Numbers
 real numbers
 Appears in these related concepts: Introduction to Domain and Range, The Intermediate Value Theorem, and Piecewise Functions
 set
 Appears in these related concepts: Sequences of Mathematical Statements, Sequences, and Introduction to Sequences
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 unknown
 Appears in these related concepts: Linear Equations and Their Applications, Equations and Inequalities, and Introduction to Variables
 variable
 Appears in these related concepts: Controlling for a Variable, Fundamentals of Statistics, and Math Review
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