Standard Form

Standard form is another way of arranging a linear equation. In the standard form, a linear equation is written as:

 $\displaystyle Ax + By = C $

where $A$ and $B$ are both not equal to zero. The equation is usually written so that $A \geq 0$, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the standard form. 

For example, consider an equation in slope-intercept form: $y = -12x +5$. In order to write this in standard form, note that we must move the term containing $x$ to the left side of the equation. We add $12x$ to both sides:

$\displaystyle y + 12x = 5$

The equation is now in standard form.

Using Standard Form to Find Zeroes

Recall that a zero is a point at which a function's value will be equal to zero ($y=0$), and is the $x$-intercept of the function. We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form. However, the zero of the equation is not immediately obvious when the linear equation is in this form. However, the zero, or $x$-intercept of a linear equation can easily be found by putting it into standard form.

For a linear equation in standard form, if $A$ is nonzero, then the $x$-intercept occurs at $x = \frac{C}{A}$. 

For example, consider the equation $y + 12x = 5$.

In this equation, the value of $A$ is 1, and the value of $C$ is 5. Therefore, the zero of the equation occurs at $x = \frac{5}{1} = 5$. The zero is the point $(5, 0)$.

Note that the $y$-intercept and slope can also be calculated using the coefficients and constant of the standard form equation. If $B$ is non-zero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\frac{C}{B}$, and the slope of the line is $-\frac{A}{B}$. 

Example: Find the zero of the equation $3(y - 2) = \frac{1}{4}x +3$

We must write the equation in standard form, $Ax + By = C$, which means getting the $x$ and $y$ terms on the left side, and the constants on the right side of the equation.

Distribute the 3 on the left side:

$\displaystyle 3y - 6 = \frac{1}{4}x +3$

Add 6 to both sides:

$\displaystyle 3y = \frac{1}{4}x + 9$

Subtract $\frac{1}{4}x$ from both sides:

$\displaystyle 3y - \frac{1}{4}x = 9$

The equation is in standard form, and we can substitute the values for $A$ and $C$ into the formula for the zero:

$\displaystyle \begin{aligned} x &= \frac{C}{A} \\&= \frac{9}{3} \\&= 3 \end{aligned}$

The zero is $(3, 0)$.