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 $ax+b=0, $ where $a$ and $b$ are real numbers and $a\neq 0$. In an equation where $x$ is a real number, the graph is the collection of all ordered pairs with any value of $y$ paired with that real number for $x$. 

For example, to graph the equation $x-1=0, $ a few of the ordered pairs would include:

• $(1,-3)$
• $(1,-2)$
• $(1,-1)$
• $(1,0)$
• $(1,1)$
• $(1,2)$
• $(1,3)$

These can also be found by solving the equation of the graph for $x$, which yields $ x = 1$. This means that the $y$-values of the points don't matter as long as their $x$-values are 1. The graph is therefore a vertical line through those points, since all points have the same $x$-value.

The same is true for an equation written as $ay+b=0$, or $y=-4$, for example. The graph would be a horizontal line through points that all have $y$-values of -4. 

Therefore, when the equation is $y=C, $ where $C$ is a constant real number, the graph is a horizontal line. Similarly, if the equation is $x=C,$ then the graph is a vertical line.

Graphs of Equations with Two Variables

The graph of a cubic polynomial has an equation like $y=x^3-9x$.  Its equation has two variables, $x$ and $y$, and the equation is solved for $y$.

Plot specific points by substituting chosen $x$-values into the equation, and solve for the corresponding $y$ value, and then graph.

Let's choose values for $x$ from -2 to 2. When $x=-2$, we have: 

$y=(-2)^3-9(-2)=(-8)+18=10$

Therefore, $(-2,10)$ is a point on this curve (i.e., the graph of the equation).

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 $y=x^3-9x$

Since the exponent of $x$ is a 3, it means that this equation is a 3rd-degree polynomial, called a cubic polynomial.