Linear equations
A linear equation is an equation that represents a straight line. When plotted on a graph, a linear equation always results in a straight line, as in the figure below, which is a special case of a linear equation referred to as the identity function, y = x:
It is referred to as the identity function because inputting any xvalue will return the same value of y.
In most cases, the term "linear equation" refers to onevariable linear equations (like the one above), though linear equations can have more variables, as long as all of the variables are firstorder variables. A firstorder variable, also referred to as a variable of degree 1, refers to the exponent on the variable. For example, the equation below is not a linear equation. The degree of each term is included for reference, though by convention, usually only exponents that are not 0 or 1 are shown:
y^{1} = 2x^{3}  3x^{1} + 4^{0}
The degree of a polynomial is determined by the highest order term in the polynomial. In this equation, the highest degree is 3, making the equation nonlinear. All variables involved must be of degree 1 for the equation to be linear.
Any number of constants can be present (since their degree is 0). If an equation only has a constant, the equation is a horizontal (ex. y = 4) or vertical line (ex. x = 2).


Forms of linear equations
Linear equations can take on multiple forms including slopeintercept form, pointslope form, standard form, and more.
Slopeintercept form
Slopeintercept form is probably the most commonly used form of a linear equation. It is typically expressed as
y = mx + b
where m is the slope, and b is the yintercept.
The equation of a line can be written in slopeintercept form when the slope and yintercept of the line are known. Or, given an equation in slopeintercept form, it is easy to quickly identify the slope and yintercept of the line. Graphing the equation of a line is also relatively simple when given an equation in slopeintercept form.
Pointslope form
Pointslope form is similar to slopeintercept form except that it is based on some point on the line, rather than the yintercept specifically. It is typically expressed as
y  y_{1} = m(x  x_{1})
where (x_{1}, y_{1}) represent a point on the line, and m is the slope. Like slopeintercept form, it is also useful for graphing, and has the benefit of being usable using any point on the line rather than just the yintercept specifically, as is the case with slopeintercept form.
Standard form
The standard form for the equation of a line is typically expressed as
Ax + By = C
Where A, B, and C are integers, and A and B are not equal to 0. One of the key benefits of standard form over slopeintercept or pointslope form is that it can be used to quickly find the xintercept. It can also be used to find the yintercept of a line, but this is something that is also relatively easy using either of the other forms. Once the xintercept and yintercept are known, standard form can also be used to graph the line, though it is slightly more tedious than using either of the other two mentioned forms.
Standard form is also the form of a linear equation that is typically used when solving systems of linear equations. Using either slopeintercept or pointslope form would make solving linear equations more difficult.
Linear equations are also expressed in other forms, but the above are some of the most common.