simultaneous linear equations

[paulo1913]

how do you solve pairs of simultaneous linear equations?

[NeonBlack]

There are many ways.

My personal favorite is Cramer's rule, which I don't have time to explain right now, maybe you can look it up on wiki.

The simplest is probably just to write your equations in the matrix form [math]Ax=B[/math] Then find the inverse matrix of A and then [math]x=A^{-1}B[/math]

[paulo1913]

Is there a way to do it which is simpler? I just don't understand that...

[Cap'n Refsmmat]

There's the substitution method.

 

[math]2x + 4y = 10[/math]

[math]2x = y[/math]

 

Your goal is to find x and y. You know that y is the same as 2x because of the second equation, so you can change the first to:

[math]2x + (4 \times 2x) = 10[/math]

or just

[math]2x + 8x = 10[/math]

Solve it and you find that x = 1.

 

Now you just take the second equation to find y.

 

It can be more complicated than that -- you may have to solve one equation for y or x to get started -- but substitution works that way in general.

[Royston]

Alternatively there's the elimination method...

 

[math]3 + 5x = 2y[/math]

[math] 5 - x = y[/math]

 

You need to find a way to make the multiple of either variable (x or y) the same in both equations, so the obvious would be to multiply by 5...

 

[math] 3 + 5x = 2y[/math]

[math] 25 - 5x = 5y[/math]

 

Add the two values on the left, the x terms cancel, so you get...

 

[math] 28 = 7y[/math] or [math]y = 4[/math]

 

Now just substitute [math]y = 4[/math] into the original equation, and you get [math]x = 1[/math]

 

I'll show you another alternative, but I'm just about to head out. Depending on the equation, some methods are more appropriate than others.