diff --git a/spec/API_specification/linear_algebra_functions.md b/spec/API_specification/linear_algebra_functions.md
index 209a2f787..729c81af6 100644
--- a/spec/API_specification/linear_algebra_functions.md
+++ b/spec/API_specification/linear_algebra_functions.md
@@ -269,10 +269,26 @@ TODO
 
 TODO
 
-(function-solve)=
-### solve()
+(function-linalg-solve)=
+### linalg.solve(x1, x2, /)
 
-TODO
+Returns the solution to the system of linear equations represented by the well-determined (i.e., full rank) linear matrix equation `AX = B`.
+
+#### Parameters
+
+-   **x1**: _<array>_
+
+    -   coefficient array `A` having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Must be of full rank (i.e., all rows or, equivalently, columns must be linearly independent). Should have a floating-point data type.
+
+-   **x2**: _<array>_
+
+    -   ordinate (or "dependent variable") array `B`. If `x2` has shape `(..., M)`, `x2` is equivalent to an array having shape `(..., M, 1)`, and `shape(x2)` must be compatible with `shape(x1)[:-1]` (see {ref}`broadcasting`). If `x2` has shape `(..., M, K)`, each column `k` defines a set of ordinate values for which to compute a solution, and `shape(x2)[:-1]` must be compatible with `shape(x1)[:-1]` (see {ref}`broadcasting`). Should have a floating-point data type.
+
+#### Returns
+
+-   **out**: _<array>_
+
+    -   an array containing the solution to the system `AX = B` for each square matrix. The returned array must have the same shape as `x2` (i.e., the array corresponding to `B`) and must have a floating-point data type determined by {ref}`type-promotion`.
 
 (function-svd)=
 ### svd()