Forms of the Equation of a Line

The functional form of a linear equation is: |y=mx + b|, where |m| is the slope of the line and |b| is its y-intercept (initial value).

The linear functions below are expressed in functional form:

|y = 2x + 3|, where |m = 2| and |b = 3|

|y = -3x - 6|, where |m = -3| and |b = -6|

|y = \frac{1}{2}x + \frac{3}{4}|, where |m = \frac{1}{2}| and |b = \frac{3}{4}|

The functional form makes it possible to express all the straight lines except the vertical lines of the form |x = constant|. In this form, the slope |m| is undefined since it is impossible to divide by 0, which is the case when calculating the slope of a vertical line.

The general form of a linear function is:

|Ax + By + C = 0|,

where

|A| and |B| must not be equal to |0| at the same time.
In general, |A,| |B|, and |C| are whole numbers and |A| is positive.

The general form of a linear equation makes it possible to express all types of straight lines, i.e., vertical, horizontal, increasing, or decreasing.

The linear function equations below are in the general form:

|2x - 3y + 7 = 0|

|x + 6y - 9 = 0|

|-3x + 4 = 0 \Rightarrow| Vertical line

|6y - 3 = 0 \Rightarrow| Horizontal line

The linear function below is expressed in a form that resembles the general form, but the coefficient of the variable |x| is not an integer and is not positive.

|\displaystyle \frac{-x}{2} + 3y - 7 = 0|

It is possible to multiply all the terms by -2 to eliminate the fraction and the parameter’s negative sign |A| to get the general form.

|\displaystyle -2\times \left(\frac{-x}{2} + 3y - 7 = 0 \right)|

The result is:

|x - 6y + 14 = 0|.

The symmetrical form of a linear equation is:

|\displaystyle \frac{x}{a}+\frac{y}{b}=1|

where
|a| is the x-intercept (zero) and
|b| is the intercept (the initial value).

The |x| and the |y| must be the only elements present in their respective fraction’s numerator.

It is also important for |a\neq 0| and |b\neq 0| since dividing by 0 is undefined.

The slope can be calculated with the following formula: |\displaystyle m=\frac{-b}{a}|.

The following linear function is in symmetrical form:

|\displaystyle \frac{x}{3}+\frac{y}{4}=1|.

The following linear function is not in symmetrical form:
|\displaystyle \frac{2x}{3}-\frac{7y}{4}=1|.

However, it is possible to express it in symmetric form by inverting the coefficients of |x| and |y| and placing them in the denominator:

|\displaystyle \frac{x}{(\frac{3}{2})}+ \frac{y}{(\frac{-​4}{7})}=1|.

The symmetrical form makes it possible to express the majority of straight lines equations with these three exceptions:

• Vertical lines (there would be no y-intercept |b|).

• Horizontal lines (there would be no x-intercept |a|).

• Straight lines passing through the origin |(0,0)| (it is impossible to divide by 0).

To switch to the general form from the functional form of the equation |\displaystyle y = \frac{4}{5}x - 4|, make the equation equal to 0 and ensure the coefficients are whole numbers.

1. Multiply the two sides of the equality by 5 to eliminate any fractions, incorporate integer coefficients, and ensure the |A| is positive. ||\begin{align}\displaystyle 5\ (y) &=5\ \left(\frac{4}{5}x-4 \right)\\5y &=4x-20 \end{align}||

2. Move the |5y| to the other side of the equation to make the whole equality zero.
   ||0 = 4x – 5y – 20||

To switch to the symmetrical form from the general form |0 = 4x – 5y – 20|, transform the equation so it is equal to |1|.

1. Move the |20| to the other side of equality.
   ||20 = 4x -5y||

2. The equality must be equal to |1|. Thus, divide the terms by |20|.
   ||\displaystyle \frac{20}{20} = \frac{4}{20}x - \frac{5}{20}y||

3. After simplifying, the result is:
   ||\displaystyle 1 = \frac{x}{5} - \frac{y}{4}||

   Thus, we learn the x-intercept of the line is |5| and its y-intercept is |\text{-}4|.

To switch to symmetrical form from functional form |\displaystyle y = \frac{4}{5}x - 4|, place the variables on the same side of the equality and ensure that the equation is equal to |1|.

1. Move the |\frac{4}{5}x| to the other side of the equality.
   ||-\dfrac{4}{5}x+y = -4||

2. The equality must be equal to |1|. Thus, divide all the terms by |\text{-}4|.
   ||\displaystyle \frac{\frac{-4}{5}x}{-4} + \frac{y}{-4} = \frac{-4}{-4}||

3. After simplifying, the result is:
   ||\displaystyle \frac{x}{5} - \frac{y}{4}=1||

To switch to the general form from the symmetrical form |\displaystyle \frac{x}{5} - \frac{y}{4}=1|, move all the terms to the same side of the equality and ensure there are no more fractions.

1. Move the |1| to the other side of the equality.
   ||\displaystyle \frac{x}{5} - \frac{y}{4}-1=0||

2. Multiply all the terms by the LCM of A and B: |\text{LCM}(5,4)=20|
   ||\begin{align}\displaystyle 20\left(\frac{x}{5}\right) + 20 \left(-\frac{y}{4}\right)+20(-1)&=20(0)\\ 4x \phantom{)+20(}-5y\phantom{+20}-20\phantom{())}&=\phantom{()}0 \end{align}||

To switch to functional form from the general form |0 = 4x – 5y – 20|, isolate the variable |y|.

1. Move the |4x| and the |20| to the other side of equality.
   ||-5y=-4x+20||

2. Divide all the terms by the coefficient of |y|.
   ||\displaystyle \frac{-5y}{-5}=\frac{-4x}{-5}+\frac{20}{-5}||

3. After simplifying, the result is:
   ||y=\dfrac{4}{5}x-4||

To switch to the functional form from the symmetrical form |\displaystyle \frac{x}{5} - \frac{y}{4}=1|, isolate the variable |y|.

1. Move the |\frac{x}{5}| to the other side of the equality.
   ||\displaystyle - \frac{y}{4}=-\frac{x}{5}+1||

2. Multiply by -4 (both sides of the equality) to isolate |y|.
   ||\begin{align}\displaystyle -4\left(- \frac{y}{4}\right)&=-4\left(-\frac{x}{5}+1\right)\\y\ \ \ \ &=\ \ \ \ \dfrac{4}{5}x\ \ -\ \ 4 \end{align}||