Why Standard Form is Useful

Turning an Equation into Standard Form

The ​slope​ is how steep our line is. In algebraic terms, it's the change in ​y​ divided by the change in ​x​. If we have two points, (​x​₁, ​y​₁) and (​x​₂, ​y​₂), the slope is:

\frac{y_2 - y_1}{x_2 - x_1}

So for our example, our points are (1,1) and (2,3) so the slope is:

\begin{aligned} \text{slope} &= \frac{3 - 1}{2 - 1} \\ \,\\ &=\frac{2}{1} = 2 \end{aligned}

Remember that ​point-slope form​ looks like this:

y - y_1 = m(x - x_1) .

​x​ and ​y​ are just our variables, but ​x​₁ and ​y​₁ are the coordinates of a specific point on the line and ​m​ is the slope.

So let's plug in the slope from our example and one of our points, (1,1), to create an equation point-slope form.

Point-slope form:

y - 1 = 2(x - 1)

Now simplify:

y - 1 = 2x - 2

​Slope-intercept form​ has this format:

y = mx + b

where ​m​ is the slope of the line and ​b​ is the ​y​-intercept.

To get from point-slope form to slope-intercept form, we want to get ​y​ by itself on the left side of the equation.

Right now we have ​y​ − 1 = 2​x​ − 2. So let's add 1 to both sides so we can get ​y​ by itself:

y = 2x − 1

When we added 1 on the left side, it canceled out with the −1. When we added 1 on the right side, we added it to the constant that was already there and got −2 + 1 = −1.

Remember that standard form looks like this:

Ax + By = C

So let's move our 2​x​ to the other side of the equals sign by subtracting 2​x​ from both sides:

-2x + y = 2

When we subtracted 2​x​ on the right side, it canceled out. When we subtracted it on the left, we put it in front of the ​y​ so it's in our pretty standard form.

So the standard form of this equation is −2​x​ + ​y​ = 2, where ​A​ = −2, ​B​ = 1 and ​C​ = 2.

Congratulations! You just turned an equation from slope-intercept form into standard form, and you learned how to write an equation in standard form using only two points.