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How to Change the Equation of a Straight Line

Here’s the general form of the straight line again, ax + by + c = 0. There are some important things to know about this general form. Firstly, the variables x and y are in alphabetical order. You should be familiar with this idea from algebra. Secondly, the coefficients a and b are integers where possible, not fractions. Thirdly, while we use plus signs in this general formula these signs may be in fact minus signs depending on the value of the coefficients. And lastly, as c is independent of both x and y and not changed by their value it’s called the constant, just as it is in algebra.

Let’s start with an equation in the normal gradient intercept form, and change it into the general form. We will use y = 3/4x – 6. How will we change this into the general form?

We want all the terms on one side and zero on the other. We also want to get rid of the fraction as a coefficient of x. This is a good time to remember your algebra skills. We can see that x has a positive coefficient of three-quarters, so we’ll place everything on that side of the equation. We’ll subtract y from both sides to move it across which gives 0 = 3/4x – y – 6. Now we need to multiply through by four to remove the fraction. This leaves us with 0 = 3x – 4y – 24. Turn that around so the zero is on the opposite side and the equation is now in general form.

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