How to find the equation of a line

If you know the intercepts.

Here is a sample problem:

you have the intercepts:

x intercept: 4
y intercept: -2

1. From here, we can find out two points:

(4, 0) and (0, -2)

Now, since we have two points, we must have two equations.

y = mx + b is the standard equation of a line.

2. Let's plug the values in for the x intercept.

y = mx + b
0 = m (4) + b
0 = 4m + b  

3. Now y intercept:

y = mx + b
-2 = m (0) + b
-2 = 0 + b
-2 = b

4. Since we found what b is in this equation, we can sub it into the first equation.

0 = m (4) + b,    b = -2     * m (4) is the same as 4m.

0 = 4m  -2
2 = 4m
2/4 = m
m = 2/4
m = 1/2
m = 0.5

5. Now we know what m and b are. We can form the equation of the line.

y = 0.5x -2

That is your answer!


Why can't you plug x and y intercept points into the same (ONE) equation? Because they change. They are not constant - x and y can be any number on the graph!

Only the values of m and b are constant. So that's why you don't write what y and x are (in numbers) in the equation!

Also, if the question asks for you to put the equation in standard form, Ax + By = C, it is very simple!

General: y = mx + b, the constants are m and b

Standard: Ax + By = C , the constants are A, B, and C

In the general equation, there is no constant infront of the y, This is because the constant that would be infront of y is included in m and b values.  The general equation likes to have y isolated, so that it is easy to see the equation. so the x and b values have already been divided by the constant that would have been infront of y.

Example:

random equation: 2y = 3x + 6

general form: y = 3/2 x + 6/2

simplified general form: y = 1.5 x + 3

Now, to change from the simplified general form into standard form, we just have to know this.

A = m

B = 1 

C = b

Let's plug the numbers in for the equation y = 1.5 x + 3

A = m = 1.5

B = 1

C = b = 3

1.5 x - y = -3  (we have to account for the rearranging of the equation)

so:

y = 1.5 x + 3

y - 1.5 x = 3,  but we want x to be positive...

1.5 x - y = -3

* if our equation is 2y = 3x + 6

A = m = 3

B = (constant infront of y) = 2

C = b = 6

Ax + By = C  plug them in

3x + 2y = 6  --> but this is not accurate, because we didn't account for the rearranging (possible sign changes)

so:

2y = 3x + 6

2y - 3x = 6

-3x + 2y = 6,  we want a positive x , so multiply both sides by negative one to cancel out the negative.

3x - 2y = -6