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Forms of Linear Equations

Forms of Linear Equations

Author: Sophia Tutorial

This lesson introduces forms of linear equations.

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  • Slope-Intercept Form
  • Slope-Point Form
  • Standard Form

Forms of Linear Equations

Linear equations can be written in several forms. Each form has its pros and cons as to why we would want to express the equation in such a format.  This is because certain information about the line and the linear relationship it represents can be easily identified just by looking at its equation.

Slope-Intercept Form

The equation of a line written in slope-intercept form is:

We refer to this form as slope-intercept form, because the equation readily gives us information about the line's slope, and its y-intercept.  The variable m represents slope, and the variable b represents the y-coordinate of the y-intercept (remember that the x-coordinate of a y-intercept is always zero). 

Here is an example of an equation written in slope intercept form:

In the second part of the example above, we used the coordinate point (0, 3) which is our y-intercept, to confirm that everything is correct.  Since 3 = 3 is a true statement, we have correctly identified 8 as the slope, and (0, 3) as the location of the y-intercept. 

Point-Slope Form

Linear equations can also come written in Point-Slope form.  Point-Slope form, as the name suggests, provides information about the lines slope, and a point on the line.  Point-Slope form is as follows:

Once again, we can easily identify the line's slope by the variable m.  Here, we also have x1 and y1.  These represent the x-coordinate and y-coordinate of a point on a line.  Below is an example of a linear equation written in Point-Slope form:

This tells us that the line has a slope of 3, and that the point (2, 7) is a point on the line.  

Be careful with equations such as (y + 3) = 2(x – 4)  Because our general form has minus signs with our x's and y's, if we see a plus sign in a specific equation, that coordinate is a negative value.  In this example, the point (–3, 4) is a point on the line. 

Standard Form

A final form we will discuss today is called Standard Form.  Unlike slope-intercept form, or point-slope form, we cannot readily identify the slope, y-intercept, or point on a line simply by looking at the equation in standard form.  However, the benefit of standard form is that any linear equation can be written in standard form, whereas not every line can be written in slope-intercept or point-slope forms. Think about a vertical line.  It is an undefined slope.  Both slope-intercept and point-slope forms rely on a defined slope to generate its equation.  A vertical line, however, can be written in standard form, because a slope is not needed to write its equation.

Here is the standard form for a linear equation:

A couple of notes about general accepted rules for equations written in standard form:

  • A, B, and C should be integers. If any of them are not, the entire equation should be multiplied so that they are, if possible.
  • A should be a positive integer.  B and C are allowed to be negative, but if A is negative, the equation should be multiplied by –1 so as to make A positive. 
  • Wherever possible, A, B, and C should be relatively prime.  This means that they should have no common factors other than 1, if possible.  For example, mathematicians prefer 3x – 2y = 6 to be written as 2x – y = 3, canceling out the common factor of 2 in both Ax, By and C. 
Formulas to Know
Point-Slope Form of a Line

y minus y subscript 1 equals m left parenthesis x minus x subscript 1 right parenthesis

Slope-Intercept Form of a Line

y equals m x plus b

Standard Form of a Line

A x plus B y equals C