diff --git a/src/gui/PivotGui.cpp b/src/gui/PivotGui.cpp
index d206049..a5e41a7 100644
--- a/src/gui/PivotGui.cpp
+++ b/src/gui/PivotGui.cpp
@@ -4,6 +4,8 @@
 #include "Solver.h"
 #include <imgui.h>
 
+static std::string equationsResultImage;
+
 static Matrix LoadMatrixFromStdVect(const std::vector<std::vector<int>>& data) {
 	Matrix result {data.size(), data.empty() ? 0 : data[0].size()};
 	for (std::size_t i = 0; i < result.GetRawCount(); i++) {
@@ -15,6 +17,9 @@ static Matrix LoadMatrixFromStdVect(const std::vector<std::vector<int>>& data) {
 }
 
 static std::string PrintVect(const Vect& vect) {
+	if	(vect.GetCardinal() == 0)
+		return "{0}";
+
 	std::string result = "Vect( ";
 	for (std::size_t i = 0; i < vect.GetCardinal(); i++) {
 		Matrix vector = vect.GetVector(i);
@@ -84,7 +89,10 @@ void PivotGui::Render() {
 		}
 	}
 
-	ImGui::Text("Matrice finale:");
+	// Display the equationsResult strings in the GUI if they are not empty
+	if (!equationsResultImage.empty()) {
+		ImGui::TextWrapped(equationsResultImage.c_str());
+	}
 
 	ImGui::EndChild();	// End Matrice initiale
 
@@ -100,7 +108,7 @@ void PivotGui::Render() {
 
 	static std::string result = "RIEN";
 
-	ImGui::Text(result.c_str());
+	ImGui::TextWrapped(result.c_str());
 
 	ImGui::End();  // End fenetre top right
 
@@ -110,22 +118,36 @@ void PivotGui::Render() {
 	ImGui::Begin("Bottom Part", nullptr,
 		ImGuiWindowFlags_NoResize | ImGuiWindowFlags_NoMove | ImGuiWindowFlags_NoTitleBar | ImGuiWindowFlags_NoScrollbar);
 
-	// rajouter des boutons clickables
-	if (ImGui::Button("Calcul noyau")) {
-		// Code de calcul
-		Vect kernel = solver.Kernel(LoadMatrixFromStdVect(matrixValues));
+	if (ImGui::Button("Calcul")) {
 
-		result = PrintVect(kernel);
-	}
-	ImGui::SameLine();	// Align buttons horizontally
-	if (ImGui::Button("Calcul rang")) {
-		// Code de calcul
-		std::size_t rank = solver.Rank(LoadMatrixFromStdVect(matrixValues));
-	}
-	ImGui::SameLine();	// Align buttons horizontally
-	if (ImGui::Button("Calcul image")) {
-		// Code de calcul
+		// Calculate the kernel and image
 		Vect image = solver.Image(LoadMatrixFromStdVect(matrixValues));
+		Matrix linearSystem = image.GetLinearSystem();
+
+		// Create a column matrix with as many elements as the number of columns in the linear system
+		std::vector<std::string> columnMatrix(linearSystem.GetColumnCount());
+		for (size_t i = 0; i < linearSystem.GetColumnCount(); ++i) {
+			columnMatrix[i] = std::string(1, 'a' + static_cast<char>(i));
+		}
+
+		// Multiply the linear system matrix by the column matrix
+		std::vector<std::string> resultMatrix(linearSystem.GetRawCount());
+		for (size_t i = 0; i < linearSystem.GetRawCount(); ++i) {
+			for (size_t j = 0; j < linearSystem.GetColumnCount(); ++j) {
+				resultMatrix[i] += std::to_string(static_cast<int>(linearSystem.at(i, j))) + "*" + columnMatrix[j] + " + ";
+			}
+			resultMatrix[i] = resultMatrix[i].substr(0, resultMatrix[i].length() - 3) + " = 0";
+		}
+
+		// Store the equationsResult strings in the global variable
+		equationsResultImage = "Equations cartesiennes de l'espace vectoriel (Image):\n";
+		for (const auto& equation : resultMatrix) {
+			equationsResultImage += equation + "\n";
+		}
+
+		result = std::string("Noyau: ") + "\n" + PrintVect(solver.Kernel(LoadMatrixFromStdVect(matrixValues))) + "\n" + "\n" +
+				 "Rang: " + "\n" + std::to_string(solver.Rank(LoadMatrixFromStdVect(matrixValues))) + "\n" + "\n" +
+				 "Image: " + "\n" + PrintVect(image);
 	}
 
 	ImGui::End();  // End fenetre bas
diff --git a/src/gui/PivotGui.h b/src/gui/PivotGui.h
index a9245fe..de534f1 100644
--- a/src/gui/PivotGui.h
+++ b/src/gui/PivotGui.h
@@ -7,4 +7,4 @@ void Init();
 void Render();
 void Destroy();
 
-}  // namespace PivotGui
+}  // namespace PivotGui
\ No newline at end of file