diff --git a/notebooks/LEQ.ipynb b/notebooks/LEQ.ipynb
index a7e4fca..03bdcda 100644
--- a/notebooks/LEQ.ipynb
+++ b/notebooks/LEQ.ipynb
@@ -2,53 +2,59 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "599913a3",
+   "id": "9c32b051",
    "metadata": {},
    "source": [
     "# Solving Linear Equations\n",
     "\n",
     "## Serial Algorithm\n",
-    "First, we construct a linear equations system $Ax=b$."
+    "To demonstrate the algorithm, we will consider a simple system of linear equations $Ax = b$:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "id": "fe3b5b02",
+   "execution_count": 66,
+   "id": "2b369b73",
    "metadata": {},
    "outputs": [],
    "source": [
-    "n = 5\n",
-    "A = rand(-10.0:10.0, (n, n))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "57c912fb",
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "x = rand(-10.0:10.0, n)\n",
-    "b = A * x"
+    "A = [1.0 4.0 5.0 8.0 1.0; \n",
+    "    2.0 -1.0 4.0 3.0 0.0; \n",
+    "    7.0 6.0 3.0 -4.0 5.0; \n",
+    "    -3.0 4.0 2.0 2.0 2.0; \n",
+    "    0.0 -4.0 2.0 1.0 2.0]\n",
+    "\n",
+    "b = [61.0; 24.0; 37.0; 29.0; 12.0];"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "4baa0681",
+   "id": "53124eb8",
    "metadata": {},
    "source": [
-    "The code in the following cell converts the problem $Ax=b$ to the upper triangular equation system $Ux=y$. "
+    "The code in the following cell converts the general problem $Ax=b$ to the upper triangular equation system $Ux=y$. Note that this function assumes that the pivots are all nonzero. This function will be erroneos if any of the diagonal entries are zero!"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "id": "11d255e0",
+   "execution_count": 67,
+   "id": "7a7b926a",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "convert_to_upper_triangular! (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 67,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "function convert_to_upper_triangular(A,b)\n",
+    "function convert_to_upper_triangular!(A,b)\n",
+    "    n = size(A,1)\n",
     "    # Upper Triangularization: convert Ax=b to Ux=y\n",
     "    for k in 1:n\n",
     "        for j in k+1:n\n",
@@ -72,20 +78,31 @@
   },
   {
    "cell_type": "markdown",
-   "id": "02c47593",
+   "id": "78ef1849",
    "metadata": {},
    "source": [
-    "The function in the following cell solves the upper triangular equation system using backwards substitution. "
+    "The function in the following cell solves the upper triangular equation system using backwards substitution. Note that the function alters the input values. "
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "id": "3c19497c",
+   "execution_count": 68,
+   "id": "5a134433",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "solve_upper_triangular! (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 68,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "function solve_upper_triangular(U,y)\n",
+    "function solve_upper_triangular!(U,y)\n",
     "    n = size(U,1)\n",
     "    for step in reverse(1:n)\n",
     "        if U[step,step] == 0\n",
@@ -108,18 +125,34 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "id": "2c24d85e",
+   "execution_count": 69,
+   "id": "b92332f7",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "5-element Vector{Float64}:\n",
+       " 1.0000000000000009\n",
+       " 1.999999999999999\n",
+       " 2.9999999999999964\n",
+       " 4.000000000000002\n",
+       " 5.000000000000005"
+      ]
+     },
+     "execution_count": 69,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "U,y = convert_to_upper_triangular(A,b)\n",
-    "sol = solve_upper_triangular(U,y)"
+    "U,y = convert_to_upper_triangular!(A,b)\n",
+    "sol = solve_upper_triangular!(U,y)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "1c356b5a",
+   "id": "962ec4a9",
    "metadata": {},
    "source": [
     "We can test if the obtained solution is correct using `@test`:"
@@ -127,19 +160,30 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "id": "5c71828d",
+   "execution_count": 70,
+   "id": "0e336c85",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
+      ]
+     },
+     "execution_count": 70,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "using Test\n",
-    "@test sol ≈ x"
+    "@test sol ≈ [1.0; 2.0; 3.0; 4.0; 5.0]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "fe3c5374",
+   "id": "28dff449",
    "metadata": {},
    "outputs": [],
    "source": []