Standard Form of a Linear Equation

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Linear equations (equations whose graphs are a line) can be written in multiple formats, but the ​standard form​ of a linear equation looks like this:

Ax + By = C

A​, ​B​ and ​C​ can be any number--including negative numbers, zero and one! So examples of standard form can look like this:

3x + 7y = 10

where ​A​ = 3, ​B​ = 7 and ​C​ = 10.

Or they can look like this:

x + 5y = 6

In this case, ​A​ = 1, ​B​ = 5 and ​C​ = 6.

Or this:

8y = 9

In this case, ​A​ = 0, which is why ​x​ does not appear in the equation. ​B​ = 8 and ​C​ = 9, as you would expect.

And here's one more:

3x − 5y = 12

Here, ​A​ = 3, ​B​ = −5 and ​C​ = 12. Notice that in this case, ​B​ is negative five!

The standard form of a linear equation is ​Ax​ + ​By​ = ​C​, where ​A​, ​B​ and ​C​ can be any number.

Why Standard Form is Useful

Standard form is great for finding the ​x​ and ​y​ intercepts​ of a graph, that is, the point where the graph crosses the ​x​-axis and the point where it crosses the ​y​-axis. Also, when solving systems of equations – finding the point where two or more functions intersect – the equations are often written in standard form.

Turning an Equation into Standard Form

You can turn an equation that's written in other formats into standard form. You can also write an equation in standard form if you're only given two points on a line, although the easiest way to do it is to go through other formats first. In this next example, we'll cover how to do both of these things: write an equation in standard form when you're only given two points, and change other equation formats into standard form.

Example: Take these two points: (1,1) and (2,3) and write the equation of the line in standard form.

We're going to go through these steps:

  1. Find the slope.
  2. Write the equation in point-slope form.
  3. Turn the equation into slope-intercept form.
  4. Turn the 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, (​x1, ​y1) and (​x2, ​y2), 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 ​x1 and ​y1 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.

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