Forms of the Linear Equation

In all mathematics and sciences, the form of presentation is extremely important. Different forms of the same equation convey different information about the relation between the variables.

In the case of linear equations -- equations that define linear paths in the Cartesian plane -- there are 4 different forms -- since it's obvious from lesson 3.2 that the point/slope form and the 2-point form are exactly the same. We covered that form -- now let's look at the other three.

We begin with the General Form -- Ax + By + C = 0, then the Standard or Slope/y-intercept Form -- y = mx + b and finally, the Symmetric Form of

Note 1: Take note of the "case" of the coefficients.

In the General Form -- always use caps for the A, B and C.

In the Standard Form -- never use caps for m and b

Note 2: Take note of the "font" of the coefficients and variables.

The standard in math textbooks and equation processors is

italics for variables as in y = m x + b -- the "m" and "b" are constants

normal font for them.

Unfortunately, as with plumbing, things are not always presented according to standard.

Note 1 is an always rule.

Note 2 is an if-we're-really-lucky rule.

The coefficients in each form indicate the properties of the line such as intercepts and slope. Since all we need to graph a line or find its equation is a slope and a point, the coefficients allow us to determine these properties by inspection.

For instance, when I look at 2x + 3y + 6 = 0, immediately I know

the x-intercept is -3, the y-intercept is -2, and the slope is

In this lesson we'll learn what the adjectives or coefficients tell us about the relation between the variables -- x's and y's -- in the linear equation.

Before we start however -- a new word for coefficients:

If you've ever listened to statisticians or politicians talk -- certainly here in Québec -- you've often heard the word "parameters". Whenever they don't know what to say about budget cuts or lack of services -- they haul that word out in phrase like:

"Well, we had to work with the parameters of the situation --

so we couldn't do everything."

They seem to think those dreaded "parameters" are a self-sustaining life force they're unable to control.

Let's look at the word.

"Para" means kind of, sort of, a type of as in para-psychology.

"Meter" as in themo-meter, baro-meter, tacho-meter means a measuring device.

So para-meters are kinds of measures.

As I said before, they indicate the properties of the line such as intercepts and slope

the " measures " we use to situate and visualize the line.

intro

General Form

Standard Form

Symmetric Form

practice & solutions

The General Form of the Linear Equation

When asked to find the equation of a line, often the question specifies in what form to write the answer.

In the previous examples, we could have left the equations in the point/slope form of

, where m is the slope.

However, we wrote all the equations, even those for lines parallel to an axis,

in the form Ax + By + C = 0.

This is the General Form of the linear equation.

Here, A is the coefficient of x, B is the coefficient of y, and C is the constant term. Remember that each of these coefficients carries its sign with it

-- signs don't float freely in the space between the terms

-- they belong to the terms that follow them.

So, if we are given the equation -5x + 3y - 7 = 0,

A = -5, B = + 3 and C = -7.

Now let's see what we can discover about the coefficients or parameters of Ax + By + C = 0.

First, we'll solve the General Equation for the x-intercept, the y-intercept and then we'll find the slope of the line this equation represents.

To solve for the x-intercept, we set y = 0 : Ax + 0 + C = 0 u Ax = -C u x = -C/A

Similarly, set x = 0 for the y-intercept: 0 + By + C = 0 u By = -C u y = -C/B

Now, we have two points on the line: , so we can find the slope.

The slope between =

Hint:

To remember which of the two is the x-intercept and which the y-intercept, since both of them have -C in the numerator, just remember that the denominator is the coefficient of the variable who's intercept you're seeking. In other words, A, which is the coefficient of x, is in the denominator of the fraction used to find the x-intercept.

Example

Find the x and y intercepts and the slope for the lines with equations:

So, as you can see, had we been asked to graph the lines in parts (a) and (b) we would simply have plotted the points (-6, 0) and (0, 10) for part (a) and (-6, 0) and (0, -2) in part (b) and we would have joined them.

In part (c) however, it would be hard to plot the y-intercept of 17/3 unless we were using teeny-tiny, hard-to-see graph paper.

But, since we know the x-intercept is (-17, 0) and we know the slope of the line is 1/3,

we could plot the point (-17, 0), rise 2 units, run 6 units and be at another point on the line.

So, knowing how to find the intercepts and slope of a line from its equation helps to graph or otherwise deal with the line.

The Standard or Slope/Y-Intercept Form of the Linear Equation

Another common and useful form of the equation of a line is the slope/y-intercept form which looks like this:

y = m x + b

where m is the slope and b is the y-intercept.

Let's solve the General Form of the equation of a line for y to show that

1. the coefficient of x is indeed the slope (-A/B)
2. the constant term is indeed the y-intercept (-C/B).

   Ax + By + C = 0

   By = -Ax - C

   y =

   When we set the equation of a line in the form y = something times x plus a constant, the something is the slope of the line and the constant is its y-intercept.

   y = mx + b is the equation of a line with slope of m and y-intercept of b.

   Examples

   Put these equations in y = mx + b form and state the slope and y-intercept

   a) 5x - 3y + 30 = 0

   b) -x - 2y - 6 = 0

   c) x - 3y + 17 = 0

   Solution

   Note: these are the same three equations we had in the last example.

   Our results will be the same.

   (a) 5x - 3y + 30 = 0 u -3y = -5x - 30 u 3y = 5x + 30 u y = + 10

   slope = ; y-int. = 10

   (b) -x - 2y - 6 = 0 u -2y = x + 6 u y = - 3

   slope = ; y-int. = -3

   (c) x - 3y + 17 = 0 u -3y = -x - 17 u 3y = x + 17 u y = x +

   slope = ; y-int. =

   Example

   Write the equation of the line with y-intercept = - 4 and slope 2/3

   Solution:

   y = x - 4

   Example

   Write the equation of the line with slope -2 through (0, 7)

   Solution:

   y = -2x + 7

   Note: Do not make the far too common mistake of believing that you need the y-intercept in order to put the linear equation into the y = mx + b form. Any point will do!

   Students often do this and it makes for more work and problems than it's worth.

   They go looking for b through a substitution step for no reason since, the point/slope form generates whatever form of the equation you want.

   Watch:

   We want the equation of the line through A(- 2, 7) with slope = 5 in Standard Form.

   Here's what I do:

   

   or y = 5x + 17, and I'm done!

   Here's the other approach:

   y = 5x + b

   7 = 5(-2) + b

   b = 17

   so y = 5x + 17

   It takes more time to look for b specifically in order to plug it into the template y = mx + b than it does to go directly to the statement that says

   the slope between any point P( x, y) and the precise point A (- 2, 7) = 5

   

   This is the equation of the line in what I call generic form.

   You see the coordinates of the defined point A and you see the slope.

   Now we perform two simple algebraic steps and we're there. We've got m and b.

   In general, you should only use this form when asked to, or in questions such as an example where you're given the slope and y-intercept or the y-intercept and another point on the line.

   In all other cases, use the point/slope form. It's precise and efficient -- exactly what we aim for in math.

   intro

   General Form

   Standard Form

   Symmetric Form

   practice & solutions

   The Symmetric Form of the Equation of a Line

   The 4th form of the linear equation is the Symmetric Form.

   This form indicates the x and y-intercepts and is written in fraction form.

   is the symmetric form of the linear equation with x-intercept a, and y-intercept b.

   Once again, we start with the General Form of the linear equation to derive the Symmetric Form. Since the right side of the Symmetric Form equation is always = 1, we divide the equation through by the constant term once its transposed like this:

   Ax + By + C = 0 u Ax + By = -C

   

   and as you can see, -C/A is the x-intercept and -C/B is the y-intercept.

   Example

   Write 3x - 5y + 15 = 0 in symmetric form.

   Solution:

   3x - 5y + 15 = 0 u 3x - 5y = -15 u when we divide through by -15.

   There will be times when the x-intercept, y-intercept or both are fractions. In such cases, the symmetric form of the equation of a line will be quite ugly, with complex fractions everywhere. However, this doesn't change the fact that when you write the symmetric equation of a line, you must write x over the x-intercept plus y over the y-intercept equals one. So you may end up with an equation that looks like this:

   

   Don't be tempted to simplify the fractions.

   Even if a or b = 1, you must write a fraction with a denominator of 1 like this:

   

   The equation must be left in this way if you were asked to write the Symmetric Form of the linear equation.

   intro

   General Form

   Standard Form

   Symmetric Form

   practice & solutions

   Practice

   1) Write this equation in standard and symmetric forms

   2x - 3y + 12 = 0

   2) Write this equation in general and symmetric forms

   

   3) Write this equation in general and standard forms

   .

   .

   Solutions

   1) standard form:

   symmetric form:

   2) general form: x - 3y - 6 = 0

   symmetric form:

   3) general form: 3x - 5y + 7 = 0

   standard form:

   .

   intro

   General Form

   Standard Form

   Symmetric Form

   practice & solutions

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