Round #1075

Solve the linear system 4x - y = 7 and 2x + 3y = 7. Give the exact values of x and y.

```csharp using System; class Program { static void Main() { // Coefficients of the linear equations double a1 = 4, b1 = -1, c1 = 7; double a2 = 2, b2 = 3, c2 = 7; // Calculate the determinant double determinant = a1 * b2 - a2 * b1; // Calculate the values of x and y double x = (b2 * c1 - b1 * c2) / determinant; double y = (a1 * c2 - a2 * c1) / determinant; Console.WriteLine("The value of x is: " + x); Console.WriteLine("The value of y is: " + y); } } ```

```csharp using System; class Program { static void Main() { // Define the coefficients of the linear system double[,] coefficients = { { 4, -1 }, { 2, 3 } }; double[] constants = { 7, 7 }; // Use the row-reduction method to solve the system double[] solution = RowReduction(coefficients, constants); // Print the solution Console.WriteLine("x = " + solution[0]); Console.WriteLine("y = " + solution[1]); } static double[] RowReduction(double[,] coefficients, double[] constants) { int n = coefficients.GetLength(0); double[,] augmentedMatrix = new double[n, n + 1]; // Create the augmented matrix for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { augmentedMatrix[i, j] = coefficients[i, j]; } augmentedMatrix[i, n] = constants[i]; } // Perform row operations to put the matrix in row-echelon form for (int i = 0; i < n; i++) { // Partial pivoting int maxRow = i; for (int k = i + 1; k < n; k++) { if (Math.Abs(augmentedMatrix[k, i]) > Math.Abs(augmentedMatrix[maxRow, i])) { maxRow = k; } } SwapRows(augmentedMatrix, i, maxRow); // Make the pivot element 1 double pivot = augmentedMatrix[i, i]; for (int j = i; j <= n; j++) { augmentedMatrix[i, j] /= pivot; } // Eliminate the pivot variable from the other rows for (int k = 0; k < n; k++) { if (k != i) { double factor = augmentedMatrix[k, i]; for (int j = i; j <= n; j++) { augmentedMatrix[k, j] -= factor * augmentedMatrix[i, j]; } } } } // Extract the solution from the augmented matrix double[] solution = new double[n]; for (int i = 0; i < n; i++) { solution[i] = augmentedMatrix[i, n]; } return solution; } static void SwapRows(double[,] matrix, int row1, int row2) { int n = matrix.GetLength(1); for (int i = 0; i < n; i++) { double temp = matrix[row1, i]; matrix[row1, i] = matrix[row2, i]; matrix[row2, i] = temp; } } } ``` This C# program uses the row-reduction method to solve the linear system 4x - y = 7 and 2x + 3y = 7. The `RowReduction` function takes the coefficients of the linear system and the constants as input, and returns the solution as an array of doubles. The `Main` function defines the coefficients and constants, calls the `RowReduction` function, and prints the solution. The row-reduction method is implemented using partial pivoting to ensure numerical stability.