Finding the intersection of two linear equations?

Question

Finding the intersection of two linear equations?

I'm looking to find a clean / elegant / not-as-clunky-as-breaking-down-almost-every-step-into-its-own-function approach towards getting Numbers to deliver the intersection of a system of two linear equations.

Setting a basic slope test to determine No Solutions (parallel) / One Solution (intersection) / Inifinite (same line) Solutions is easy enough, but I've been fumbling around for a clean way to calculate x and y for cases of One Solution.

Essentially, I'm looking for something less explicit than having a separate function for each step of solving by hand, but more explicit than Excel's equation solver. This would allow me to study and explain the basic patterns involved, and generate graphing problems automatically (with a built-in self-check).

MacPro 8 Core/8GB, Mac OS X (10.6.6), shiny

Posted on Mar 15, 2011 2:23 AM

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Answer 1

KOENIG Yvan

Level 8

41,845 points

Mar 15, 2011 2:45 AM in response to Chad Makaio Zichterman

Look at your mathematic curse.

equation 1
y = a x + b
equation 2
y = c x + d

If there is a soluce the value y is the same at the intersection so we may write

a x + b = c x + d
(a - c) x = d - b
x = (d - b) / (a - c)
y = (a * (d - b) / (a - c)) + d

Yvan KOENIG (VALLAURIS, France) mardi 15 mars 2011 10:44:26

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Answer 2

Badunit

Level 7

31,228 points

Mar 15, 2011 7:20 AM in response to KOENIG Yvan

Using Yvan's formulas,

No solution: A=C, B ≠ D
Infinite solutions: A=C, B=D
One solution: Calculated by the formulas provided by Yvan

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Answer 3

KOENIG Yvan

Level 8

41,845 points

Mar 15, 2011 7:32 AM in response to Badunit

I responded to :

I've been fumbling around for a clean way to calculate x and y for cases of One Solution.

I guess that the OP was aware of the two particular cases 😉

But when I studied, the case A=C, B ≠ D
wasn't described as no solution but as "the intersection is at the infinite" which was defining parallels.

Yvan KOENIG (VALLAURIS, France) mardi 15 mars 2011 15:27:49

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Answer 4

Mar 15, 2011 9:34 AM in response to KOENIG Yvan

KOENIG Yvan wrote:
Look at your mathematic curse.

Freudian slip, perhaps? ;p

{quote}

If there is a soluce the value y is the same at the intersection so we may write

a x + b = c x + d
(a - c) x = d - b
+x = (d - b) / (a - c)+

+> y = (a * (d - b) / (a - c)) + d+

Yvan KOENIG (VALLAURIS, France) mardi 15 mars 2011 10:44:26

{quote}

Those x and y formulas work perfectly...exactly what I was looking for.

*Side note: I feel almost as much relief at having asked a question not found in the User Guide or Formulas and Functions help...as having received a successful solution to the question itself.

Thank you both.

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