# ApCoCoA-1:LinAlg.Solve

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## LinAlg.Solve

Solves a system of linear equations.

### Syntax

```LinAlg.Solve(M:MAT, B:MAT, BACKEND:STRING):MAT
LinAlg.Solve(M:MAT, B:MAT, BACKEND:STRING, METHOD:STRING):MAT
```

### Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Let M and B be matrices defined over the ring of integers, a finite field or the field of rationals. This function tries to solve the linear equation system M*X = B by using the ApCoCoAServer supported by the IML or LinBox library (depending on the value of parameter BACKEND). Possible values for BACKEND are IML and LINBOX. If BACKEND is set to LINBOX, a fourth optional parameter METHOD may be specified. Please refer to LinBox.Solve for further details about this parameter.

The return value will be a solution vector of the linear equation system or an empty matrix if no solution has been found.

• @param M A matrix with components either of type INT, ZMOD or RAT.

• @param B A matrix with components either of type INT, ZMOD or RAT.

• @param BACKEND Either the string "IML" or "LINBOX".

• @return A matrix X representing a solution vector of the linear equation system M*X = B if a solution exists or the empty matrix otherwise.

The following parameter is optional.

#### Example

```Use ZZ/(19)[x];
M := BringIn(Mat([[1,3,4], [0,2,1]]));
B := BringIn(Mat([, ]));
LinAlg.Solve(M, B, "IML");

-------------------------------
Mat([
[-2 % 19],
[1 % 19],
[0 % 19]
])
-------------------------------
```

#### Example

```Use QQ[x];
M := Mat([ [1,3,4], [0,2,1], [1,3,4] ]);
B := Mat([ , ,  ]);
LinAlg.Solve(M, B, "IML");

-------------------------------
Mat([
[ ]
])
-------------------------------
```