A system of equations with two different lines written in the form Ax + By = 0 will always, sometimes, or never have (0, 0) as the solution.

Answer 1

A systems of equations with two different lines written in the form Ax + By = 0 will always have the solution (0,0).

Why?

The solution to a systems of equations is the point where the two lines intersect. If the solution to a systems of equations is (0,0), that means both lines pass through this point. That also means that the y-intercept of both lines is 0 (because lines only have 1 y-intercept and the point (0,0) is on the y axis). So, another way to ask this question is "do all lines in the form Ax + By = 0 have a y-intercept of 0?"

Let's find out:

Ax + By = C is the standard form for writing linear equations. If we convert it into slope-intercept form (y = mx + b), we can easily tell what the y-intercept is. So, all we need to do to answer the question is rewrite our original equation in slope-intercept form.

Ax + By = 0 (Original equation)

By = 0 - Ax (Subtract Ax from both sides)

y = 0/B - Ax/B (Divide both sides by B to isolate y)

y = - Ax + 0 (Reduce because anything/0 = 0)

Our equation, in slope-intercept form, is y = - Ax + 0. The "b" term, or the y-intercept, is 0. This means that the y-intercept for these equations is always 0. If all equations have a y-intercept of 0, they all pass through (0,0). This means if you have two of them in a systems of equations, the solution to the system will always be (0,0).