Equations of the Line

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Lines appear everywhere and can be described in lots of different ways. We always describe lines as being straight, which means that they do not deviate from their direction, and continue for ever.

In mathematics, we have several ways of describing lines using equations. We call these linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.

Contents 1 Linear equations in two variables 1.1 Forms for Linear Equations 1.1.1 General form 1.1.2 Standard form 1.1.3 Slope–intercept form 1.1.4 Point–slope form 1.1.5 Two-point form 1.1.6 Intercept form 1.1.7 Special cases 2 Changing from one form to Slope-Intercept

Linear equations in two variables

A common form of a linear equation in the two variables x and y is

where m is known as the slope of the line (or steepness) and b is the y-intercept of the line (or where i crosses the y-axis). The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line.

Forms for Linear Equations

Linear equations can be rewritten using the laws of algebra into several different forms. These equations are often referred to as the "equations of the straight line". In what follows, x and y are known as variables; other letters represent constants (or non-changing and fixed numbers).

General form

where A and B and C are constants (some of which may be zero, but not all of them at the same time). The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is the x-coordinate of the point where the graph crosses the x-axis (y is zero), is −C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (x is zero), is −C/B, and the slope of the line is −A/B.

Standard form

where A, B, and C are integers whose greatest common factor is 1, A and B are not both equal to zero, and A is non-negative (and if A = 0 then B has to be positive). The standard form can be converted to the general form, but not always to all the other forms if A or B is zero.

Slope–intercept form

where m is the slope of the line and b is the y-intercept, which is the y-coordinate of the point where the line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. Vertical lines, having undefined slope, cannot be represented by this form.

Point–slope form

where m is the slope of the line and (x₁,y₁) is any point on the line. The point-slope and slope-intercept forms are easily interchangeable.

The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, ) is proportional to the difference in the x coordinate (that is, ). The proportionality constant is m (the slope of the line).

Two-point form

where (x₁,y₁) and (x₂,y₂) are two points on the line with x₂ ≠ x₁. This is equivalent to the point-slope form above, where the slope is explicitly given as (y₂−y₁) / (x₂−x₁).

Intercept form

where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form can be converted to the standard form by setting A = 1/a, B = 1/b and C = 1.

Special cases

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.

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.

and

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.

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) and we say that the original equation has no solution. An example would be 3x + 2 = 3x − 5.

Changing from one form to Slope-Intercept

In the majority of cases that apply to an Algebra I course, the simplest way to graph a line is by looking at the slope-intercept form (y = m x + b). Most texts and problems that you will be asked to do in homework assignments and on quizzes or tests will be in either Standard Form (Ax + By = C) or in Point-Slope Form, or y - y₁ = m (x - x₁).

It is easiest at this point to look at some examples. First, let's examine a problem asking us to convert from Point-Slope to Slope-Intercept.

1) Rewrite the equation in Slope-Intercept form.

tells us that 2 is the slope and the point the line goes through is (-1, 3). Let's distribute the 2 on the right hand side.

is the result after we distribute. Now let's get the y by itself by adding 3 to both sides.

is what we end up with. Note that the +3 we did to both sides has to be combined with the +2 since that is the like term.

2) Rewrite the equation in Slope-Intercept form.

is what we start with, with 2/3 as the slope and (6, -3) as the point.

is left after we distribute. Notice that 2/3 of 6 is 4?

is the answer, after we subtract 3 from both sides.

Now let's examine a few problems that require being changed from Standard Form to Slope-Intercept.

3) Rewrite the equation into Slope-Intercept form.

is what we start with. Let's get the y by itself.

is what we have after we subtract the 2x term from both sides. Note that the 9 is positive, so we write it as + 9.

is what we have after we divide each term, both on the left and right, by 3.

4) Rewrite the equation into Slope-Intercept form.

is the original. Subtract the 2x from both sides...

is what we have next... note that the y is still negative. Divide each term (both sides) by a -1.

is the result, and in this case... our answer!