A linear equation in standard form is an equation that looks like
ax + by = c
where a, b, and c are real numbers and a and b aren't both zero. But c can be zero if it wants. It's the favorite child, so it gets special privileges.
If only a = 0, the equation can be rewritten to look like this:
y = (some number)
If only b = 0, the equation can be rewritten, too:
x = (some number)
For example, the equation 8y = 3 is equivalent to the equation 
, which is also in standard form (with b = 1).
Meanwhile, the equation 2x = 4 is equivalent to the equation x = 2, which is also in standard form (with a = 1).
If either a or b is zero, we know how to graph the equation and how to read off an equation from a graph. You probably suspect there will be some cases where it won't be so easy, and neither a nor b will be zero. You suspect right.
Okay, now what if an equation throws us a curveball? Should we sacrifice our bodies and take our base?
If neither a nor b is zero, we can most easily graph the linear equation by finding its intercepts.
Sample Problem
Graph the linear equation x + 4y = 8.
Let's find the intercepts. To find the x-intercept, let y = 0 and solve for x, since the x-intercept will be at a point of the form (something, 0).
x + 4(0) = 8
So x = 8 is the x-intercept.
For the y-intercept, let x = 0 and solve for y.
0 + 4y = 8
And y = 2 is the y-intercept. Sweet, we've tracked down both intercepts. Who needs a or b to be zero? Not us.
Now we can plot the intercepts:
Connect the dots to get the line:
Sample Problem
Write, in standard form, the linear equation graphed below:
The x intercept is at (-1, 0), which means whatever a, b, and c are, our equation looks like this:
a(-1) + b(0) = c
Let's make life easy on ourselves and let a = 1. That's right...we're going to dip this equation in a bucket of A-1 sauce.
1(-1) + b(0) = c
-1 = c
To find b, the remaining coefficient, we look at the y-intercept: y = -2. At that point, x will be 0, and we've already decided that c = -1, so we find:
0 + b(-2) = -1
Therefore, 
. We now know all the coefficients and can write the equation:
If we want to make things pretty, we can multiply both sides of the equation by 2 and write the resulting equation, which has integer coefficients. If we want to make things really pretty, we can dress the equation up in a sequined ball gown and give it a makeover. Let's start small, though:
2x + y = -2
Sample Problem
Write, in standard form, the linear equation graphed below:
The x intercept is -2, which means whatever a, b, and c are, our standard-form equation is:
a(-2) + b(0) = c
We can let a = 1, so:
-2 = c
To find b we look at the y-intercept, which occurs at (0, 4). And since we've decided c = -2, we find:
0 + b(4) = -2
This means 
. We now know all the coefficients. Not on a first-name basis, but well enough to get by. We can now write the equation.
To make things pretty, we can multiply both sides of the equation by 2 to get an equivalent equation with integer coefficients:
2x – y = -4
Now for that makeover.