# How do you write an equation in standard form for a line passing through R(3, 3), S(-6, -6)?

May 8, 2015

Method 1.
Notice that the values of X-coordinate and Y-coordinate are equal in two different points.
So, an intelligent guess would be to consider an equation
$y = x$
as the representation of this line.

Method 2.
Assume that the equation we are looking for is
$y = a \cdot x + b$
where $a$ and $b$ are unknown coefficients that we have to determine using the data provided.
Substitute the given coordinates into this equation:
$3 = a \cdot 3 + b$
$- 6 = a \cdot \left(- 6\right) + b$

Resolve the first equation for $b$ in terms of $a$ and substitute into the second equation to have only one equation with one unknown $a$:
$b = 3 - 3 a$
$- 6 = - 6 a + \left(3 - 3 a\right)$

From the latter we derive:
$- 9 = - 6 a - 3 a$
$- 9 = - 9 a$
$a = 1$

Now find $b$ that we expressed in terms of $a$:
$b = 3 - 3 a = 3 - 3 \cdot 1 = 0$

The above resulted in the equation for our line:
$y = 1 \cdot x + 0$ or, simply, $y = x$.

Method 3.
Again, looking for an equation in a format
$y = a \cdot x + b$
As it's known, the coefficient $a$ is a slope of a line that can be calculated as a ratio between increment of ordinate (Y-coordinate changes from $- 6$ to $3$) to increment of abscissa (X-coordinate changes from $- 6$ to $3$)).
In our case this ratio equal to:
$a = \frac{3 - \left(- 6\right)}{3 - \left(- 6\right)} = 1$

To determine coefficient $b$, we can substitute the values of abscissa and ordinate of one of the points to get an equation for $b$:
$3 = 1 \cdot 3 + b$
from which follows that $b = 0$.
So, our equation is
$y = 1 \cdot x + 0$ or, simply, $y = x$.