diff --git a/src/main/java/com/thealgorithms/matrix/SolveSystem.java b/src/main/java/com/thealgorithms/matrix/SolveSystem.java
new file mode 100644
index 000000000000..9e683bc4dc5c
--- /dev/null
+++ b/src/main/java/com/thealgorithms/matrix/SolveSystem.java
@@ -0,0 +1,71 @@
+package com.thealgorithms.matrix;
+
+/**
+ * This class implements an algorithm for solving a system of equations of the form Ax=b using gaussian elimination and back substitution.
+ *
+ * @link <a href="https://en.wikipedia.org/wiki/Gaussian_elimination">Gaussian Elimination Wiki</a>
+ * @see InverseOfMatrix finds the full of inverse of a matrice, but is not required to solve a system.
+ */
+public final class SolveSystem {
+    private SolveSystem() {
+    }
+
+    /**
+     * Problem: Given a matrix A and vector b, solve the linear system Ax = b for the vector x.\
+     * <p>
+     * <b>This OVERWRITES the input matrix to save on memory</b>
+     *
+     * @param matrix    - a square matrix of doubles
+     * @param constants - an array of constant
+     * @return solutions
+     */
+    public static double[] solveSystem(double[][] matrix, double[] constants) {
+        final double tol = 0.00000001; // tolerance for round off
+        for (int k = 0; k < matrix.length - 1; k++) {
+            // find the largest value in column (to avoid zero pivots)
+            double maxVal = Math.abs(matrix[k][k]);
+            int maxIdx = k;
+            for (int j = k + 1; j < matrix.length; j++) {
+                if (Math.abs(matrix[j][k]) > maxVal) {
+                    maxVal = matrix[j][k];
+                    maxIdx = j;
+                }
+            }
+            if (Math.abs(maxVal) < tol) {
+                // hope the matrix works out
+                continue;
+            }
+            // swap rows
+            double[] temp = matrix[k];
+            matrix[k] = matrix[maxIdx];
+            matrix[maxIdx] = temp;
+            double tempConst = constants[k];
+            constants[k] = constants[maxIdx];
+            constants[maxIdx] = tempConst;
+            for (int i = k + 1; i < matrix.length; i++) {
+                // compute multipliers and save them in the column
+                matrix[i][k] /= matrix[k][k];
+                for (int j = k + 1; j < matrix.length; j++) {
+                    matrix[i][j] -= matrix[i][k] * matrix[k][j];
+                }
+                constants[i] -= matrix[i][k] * constants[k];
+            }
+        }
+        // back substitution
+        double[] x = new double[constants.length];
+        System.arraycopy(constants, 0, x, 0, constants.length);
+        for (int i = matrix.length - 1; i >= 0; i--) {
+            double sum = 0;
+            for (int j = i + 1; j < matrix.length; j++) {
+                sum += matrix[i][j] * x[j];
+            }
+            x[i] = constants[i] - sum;
+            if (Math.abs(matrix[i][i]) > tol) {
+                x[i] /= matrix[i][i];
+            } else {
+                throw new IllegalArgumentException("Matrix was found to be singular");
+            }
+        }
+        return x;
+    }
+}
diff --git a/src/test/java/com/thealgorithms/matrix/SolveSystemTest.java b/src/test/java/com/thealgorithms/matrix/SolveSystemTest.java
new file mode 100644
index 000000000000..c8d289bd8339
--- /dev/null
+++ b/src/test/java/com/thealgorithms/matrix/SolveSystemTest.java
@@ -0,0 +1,21 @@
+package com.thealgorithms.matrix;
+
+import static org.junit.jupiter.api.Assertions.assertArrayEquals;
+
+import java.util.stream.Stream;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.Arguments;
+import org.junit.jupiter.params.provider.MethodSource;
+
+class SolveSystemTest {
+
+    @ParameterizedTest
+    @MethodSource({"matrixGenerator"})
+    void solveSystem(double[][] matrix, double[] constants, double[] solution) {
+        double[] expected = SolveSystem.solveSystem(matrix, constants);
+        assertArrayEquals(expected, solution, 1.0E-10, "Solution does not match expected");
+    }
+    private static Stream<Arguments> matrixGenerator() {
+        return Stream.of(Arguments.of(new double[][] {{-5, 8, -4}, {0, 6, 3}, {0, 0, -4}}, new double[] {38, -9, 20}, new double[] {-2, 1, -5}), Arguments.of(new double[][] {{-2, -1, -1}, {3, 4, 1}, {3, 6, 5}}, new double[] {-11, 19, 43}, new double[] {2, 2, 5}));
+    }
+}