Linear Equations (General and Standard forms: From Wikipedia)

[jaja1990]

Source: http://en.wikipedia.org/wiki/Linear_equation

General form:-

It says (under the title General Form) "If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is −C/B, and the slope of the line is −A/B."

How is the slope deduced to be -A/B?


Standard form:-

Aren't this form contradictory to the first form?
I mean, both are the same, but here C is positive, and on the left side of the equation, while in the General Form, C is positive, and on the right side of the equation.

 

[I like Serena]

Source: http://en.wikipedia.org/wiki/Linear_equation

General form:-

It says (under the title General Form) "If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is −C/B, and the slope of the line is −A/B."

How is the slope deduced to be -A/B?Standard form:-

Aren't this form contradictory to the first form?
I mean, both are the same, but here C is positive, and on the left side of the equation, while in the General Form, C is positive, and on the right side of the equation.

The slope of a line is the amount that y changes if x increases by 1.
If x increases by 1, Ax increases by A, so By has to decrease by A.
More specifically if y changes by -A/B, then By changes by -A.

The second form is a new equation in which the old A, B and C no longer apply.
It is more or less related to the first form, but as you already surmised the C in the second form would have to be minus the C of the first form, if A and B are the same.

 

[jaja1990]

But I thought the change doesn't have to be 1 in x? What if the change is p?

 

[I like Serena]

But I thought the change doesn't have to be 1 in x? What if the change is p?

The slope of a line is defined to be the change in y if x changes by 1.

If x changes by p, then y will change by p times the slope.

 

[jaja1990]

From Wikipedia: "The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. "

That's what I meant.

"The slope of a line is the amount that y changes if x increases by 1.
If x increases by 1, Ax increases by A, so By has to decrease by A."

So, if x increases by p, Ax increase by Ap, so By has to decrease by Ap. (So, y = Ap/B).

This means -Ap/B should be the slope in general for the equation Ax + By + C = 0. Or is my conclusion wrong?

 

[I like Serena]

As you say, the slope is the change in the y coordinate divided by the corresponding change in the x coordinate.

Since -Ap/B is the change in y if x changes by p, that means that the slope is -Ap/B divided by p, which is -A/B.

 

[jaja1990]

That's right.

Thank you :)

 

[I like Serena]

You're welcome.