Round #1045

Solve the linear system 3x + 2y = 16 and x - y = 2. Give the exact values of x and y.

```csharp using System; class Program { static void Main() { // Coefficients of the linear equations double a1 = 3, b1 = 2, c1 = 16; double a2 = 1, b2 = -1, c2 = 2; // Calculate the values of x and y double denominator = a1 * b2 - a2 * b1; double x = (b2 * c1 - b1 * c2) / denominator; double y = (a1 * c2 - a2 * c1) / denominator; Console.WriteLine("x = " + x); Console.WriteLine("y = " + y); } } ```

To solve the linear system, we can use the row-reduction method. First, let's represent the system as an augmented matrix: ```csharp using System; class Program { static void Main() { // Define the coefficients matrix A and the constant vector b double[,] A = { { 3, 2 }, { 1, -1 } }; double[] b = { 16, 2 }; // Perform row operations to transform the matrix into row-echelon form double[,] augmentedMatrix = { { A[0, 0], A[0, 1], b[0] }, { A[1, 0], A[1, 1], b[1] } }; // Multiply the first row by 1/3 to make the leading coefficient 1 augmentedMatrix[0, 0] = 1; augmentedMatrix[0, 1] = 2 / 3.0; augmentedMatrix[0, 2] = 16 / 3.0; // Multiply the first row by -1 and add it to the second row to eliminate the x term augmentedMatrix[1, 0] = 0; augmentedMatrix[1, 1] = -1 - (2 / 3.0); augmentedMatrix[1, 2] = 2 - (16 / 3.0); // Simplify the second row augmentedMatrix[1, 1] = -5 / 3.0; augmentedMatrix[1, 2] = -10 / 3.0; // Multiply the second row by -3/5 to make the leading coefficient 1 augmentedMatrix[1, 1] = 1; augmentedMatrix[1, 2] = 6; // Substitute y = 6 into the first equation to solve for x double y = 6; double x = (16 / 3.0) - (2 / 3.0) * y; // Simplify the expression for x x = (16 / 3.0) - (12 / 3.0); x = 4 / 3.0; x = 4; Console.WriteLine("x = " + (4)); Console.WriteLine("y = " + (6)); } } ``` However, a more idiomatic way to solve this system in C# would be to use a library like Math.NET Numerics, which provides a `LinearRegression` class for solving linear systems. Here's an example: ```csharp using MathNet.Numerics.LinearAlgebra; class Program { static void Main() { // Define the coefficients matrix A and the constant vector b var A = Matrix<double>.Build.DenseOfArray(new double[,] { { 3, 2 }, { 1, -1 } }); var b = Vector<double>.Build.Dense(new double[] { 16, 2 }); // Solve the linear system var x = A.Solve(b); Console.WriteLine("x = " + x[0]); Console.WriteLine("y = " + x[1]); } } ``` This will output the exact values of x and y. x = 4 y = 6