solving linear equations

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= + ÷ ± ∆ ∞ × π Σ f(x) % ≤ ! ≠ *

Given the equations of two lines, how can I find the intersection?

First, let's set up some criteria: we will be talking about two-variable circumstances. That means that when graphed, these equations will be lines graphed in a plane. (If you want to review more about lines , try equation of a line or linear or a powerpoint slide about lines.)

So, one of three conditions can exist: 1) the lines will intersect at exactly one point or 2) the lines are parallel and will not intersect, i.e. "no solution" or 3) the lines coincide (same line) and have infinite solutions. Let's look at these one at a time:

1) the lines intersect, at exactly one point. Lines graphed as we have described do not bend so once they cross they will not cross again. We use techniques such as substitution or linear combination (elimination) in order to solve these.

2) the lines are parallel and do not intersect. These lines will have the same slope but different intercepts.

3) the lines coincide, they have the same points. These lines (actually just one line) have the same slope and also the same intercepts because they have the same points.

See linear combination or substitution

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So, one of three conditions can exist: 1) the lines will intersect at exactly one point or 2) the lines are parallel and will not intersect, i.e. "no solution" or 3) the lines coincide (same line) and have infinite solutions. Let's look at these one at a time:
1) the lines intersect, at exactly one point. Lines graphed as we have described do not bend so once they cross they will not cross again. We use techniques such as substitution or linear combination (elimination) in order to solve these.
2) the lines are parallel and do not intersect. These lines will have the same slope but different intercepts.
3) the lines coincide, they have the same points. These lines (actually just one line) have the same slope and also the same intercepts because they have the same points.