# numpy.linalg.solve¶

`numpy.linalg.``solve`(a, b)[source]

Solve a linear matrix equation, or system of linear scalar equations.

Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b.

Parameters
a(…, M, M) array_like

Coefficient matrix.

b{(…, M,), (…, M, K)}, array_like

Ordinate or “dependent variable” values.

Returns
x{(…, M,), (…, M, K)} ndarray

Solution to the system a x = b. Returned shape is identical to b.

Raises
LinAlgError

If a is singular or not square.

Notes

New in version 1.8.0.

Broadcasting rules apply, see the `numpy.linalg` documentation for details.

The solutions are computed using LAPACK routine `_gesv`.

a must be square and of full-rank, i.e., all rows (or, equivalently, columns) must be linearly independent; if either is not true, use `lstsq` for the least-squares best “solution” of the system/equation.

References

1

G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 22.

Examples

Solve the system of equations `3 * x0 + x1 = 9` and `x0 + 2 * x1 = 8`:

```>>> a = np.array([[3,1], [1,2]])
>>> b = np.array([9,8])
>>> x = np.linalg.solve(a, b)
>>> x
array([2.,  3.])
```

Check that the solution is correct:

```>>> np.allclose(np.dot(a, x), b)
True
```

numpy.trace

#### Next topic

numpy.linalg.tensorsolve