Graph Linear Equation Two Variables

Special cases

y = b

This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope M = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.

x = a

This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to a. The slope is undefined. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept.

y = y & x = x

In this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity and one would not normally consider its graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y.

e = f

In situations where algebraic manipulation leads to a statement such as 1 = 0, then the original equation is called inconsistent, meaning it is untrue for any values of x and y (i.e. its graph would be the empty set ) An example would be 3x+ 2 = 3x - 5.

Connection with linear functions and operators

In all of the named forms above (assuming the graph is not a vertical line), the variable y is a function of x, and the graph of this function is the graph of the equation. In the particular case that the line crosses through the origin, if the linear equation is written in the form y = f(x) then f has the properties:

f (x+y) = f(x) + f(y)

and

f(ax) = af(x),

where a is any scalar. A function which satisfies these properties is called a linear function, or more generally a linear map. This property makes linear equations particularly easy to solve and reason about.

Note that not all linear equations are linear functions. A linear function preserves the rules of scalars. In other words, equations that have a zero y-intercept are also linear functions, but equations that have non-zero y-intercepts are not due to the fact that there is no possible way to write a matrix scalar to form the accepted slope form.