Solving 12 equations with 13 variables

$16a + 23b + 12c + 34d -2e -37f + 109g -141h + 139i + 149j + 29k + 12l + 131m = 74608$

$13a + 31b -5c + 17d + 29e -67f -16g -101h -7i -201j + 32k + 17l + 171m = -4194$

$81a -12b -5c + 17d -9e + 17f + 39g + 49h -67i -7j -41k + 13l + 121m = -15793$

$91a -43b + 17c + 19d + 17e + 23f + 97g + 101h -101i + 173j -12k + 137l + 217m = -16661$

$71a + 47b + 37c + 37d + 41e -37f -67g -17h -2i -6j + 41k -39l -12m = -6862$

$61a + 13b + 101c + 7d + 117e + 111f -4g -3h -19i -2j + 13k + 17l -91m = 7802$

$51a + 31b + 37c + 4d + 101e -13f -7g -13h -41i -3j + 31k + 73l -71m = 8846$

$21a + 12b + 67c -67d -9e -67f + 9g + 98h + 101i + 171j + 13k + 191l -54m = -4544$

$41a -2b + 5c + 17d + 18e + 19f + 20g + 21h + 22i + 23j -11k + 91l -2m = -4050$

$31a + 100b + 2c + 4d + 5e + 10f -10g + 20h -50i -4j + 10k -67l -4m = -4459$

$17a + 67b + 2c -2d -8e -78f -9g -101h + 107i + 145j + 21k -9l -10m = 37104$

$11a + 3b + 4c + 71d -45e + 87f + 8g + 34h + 45i + 93j + 104k -8l -20m = 4737$

$19a -3b -9c -12d + 123e + 61f + 71g + 83h + 91i + 10j + 11k -12l + 82m = 623$

How would I go about finding $a, b, c, d, e ... m$? I can use programming.

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Gaussian elimination? How many equations and how many unknowns do you have? –  timur Jan 9 '11 at 4:03
en.wikipedia.org/wiki/System_of_linear_equations –  user2468 Jan 9 '11 at 4:43
If you have access to linear algebra package, you could put n coefficients of each equation into a row of a matrix, say m-by-n matrix A, values on right-hand-side into m-by-1 vector b, result will be elements of A^(-1)b –  Yaroslav Bulatov Jan 20 '11 at 18:48

Have you heard of Gauss-Jordan elimination? It would be an appropriate algorithm to use for this system of linear equations.

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That, and I suggest using $x_1,x_2,x_3$ etc instead of $a,b,c$ etc. –  timur Jan 9 '11 at 4:07
Thank you so much! –  SuprDewd Jan 9 '11 at 5:08

Since you have 12 equations and 13 variables, you may have an overconstrained system, requiring not an exact solution, but a least squares solution. To compute the least-squares solution to

$A\vec{x} = \vec{b}$

where the system is overconstrained, then the least squares solution $\vec{x}^*$ is found:

$\vec{x}^* = (A^{T}A)^{-1} A^{T}\ \vec{b}$

This comes from various linear algebra concepts, specifically the projection of a vector onto a plane. More information can be found at Paul's Notes on Least Squares.

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I think you mean underconstrained (or, underdetermined) since he has more variables then equations. –  rcollyer Mar 30 '11 at 20:02