Converting Slope-Intercept Form to Standard Form

In this post, I will explain to you how to go about converting slope-intercept form to standard form when working with an equation of a line.  Expressing an equation of a line in either slope-intercept form or in standard form will both describe the exact same line, despite the equations possibly looking different.  Furthermore, sometimes in your math problems you will be asked to provide your answer in one form or the other.  Therefore, it is important for you to know how to manipulate and rearrange the expression to convert slope-intercept form to standard form.

I will explain how to obtain the equation of a line in a different post, as well as a more detailed explanation of both slope-intercept form and standard form.

Slope-intercept form is the common way of expressing an equation of a line, and it takes the general form:

y = mx + b

You may be more familiar with seeing these equations expressed as point-slope form, which is closely related to slope-intercept form. In slope-intercept form, you can see your variables x and y, and the other two values (coefficents) are the slope (m) and the intercept (b). Point-slope form looks a bit different, requiring the slope (m) and a single ordered pair for a point on the line (x₁ and y₁), as well as the variables x and y:

(y - y₁) = m(x - x₁)

Plugging values into this equation, you will find it quite simple to rearrange it to obtain the slope-intercept form of the equation.

Standard form looks a bit different still, where the x and y variables are written on the same side, and A, B, and C are all coefficients:

Ax + By = C

The general strategy of converting slope-intercept form to standard form (or by extension, converting point-slope form to standard form) is to combine and simplify to rearrange the equation so that your x and y variables (with their coefficients A and B) are on one side, and the constant value (C, the terms without an x or a y in them) on the other side of the equals sign.

Here is an example of this type of question you might see.

Express y = 10x - 20 in standard form, and state the values for A, B, and C.

y = 10x - 20 (rearrange so x and y are on the same side)

10x - y = 20
A = 10, B = (-1), C = 20

It is important to note, as in this example, the value for B is negative! This is because the standard form has a PLUS in it. Pay close attention to the sign in your answer! So, more explicitly, the standard form of this equation could be seen as 10x + (-y) = 20.

Another point to consider is that during your simplification of the answer, you may end up with fraction coefficients. In this case, it is smart to multiply everything (both sides!) by the value in the denominator (to remove it from the denominator). Mathematically, if you are doing the same thing to both sides, you aren't really changing anything, which is what makes this allowed. Keeping all numbers as whole numbers and avoiding fractions where possible is a common practice.

If you want to go the other way, and convert from standard form to point-slope form or slope-intercept form, it is basically just the reverse of what we did here.  Try it with the above example to see for yourself.  Leave me a comment below if you would like additional examples or clarifications.  Please +1 this post below if it was helpful.