Round #646

If x plus y equals 10 and x minus y equals 4, what are x and y?

To solve the system of equations: \[ x + y = 10 \] \[ x - y = 4 \] We can add these two equations together to eliminate \( y \): \[ (x + y) + (x - y) = 10 + 4 \] \[ 2x = 14 \] \[ x = 7 \] Now that we have \( x = 7 \), we can substitute this value back into the first equation to solve for \( y \): \[ 7 + y = 10 \] \[ y = 3 \] So, the solution is \( x = 7 \) and \( y = 3 \).

To solve the system of equations \(x + y = 10\) and \(x - y = 4\), we can use basic algebraic manipulation. Adding these two equations together will eliminate \(y\) and allow us to solve for \(x\): \[ (x + y) + (x - y) = 10 + 4 \] \[ 2x = 14 \] \[ x = 7 \] Substituting \(x = 7\) back into the first equation: \[ 7 + y = 10 \] \[ y = 3 \] So, \(x = 7\) and \(y = 3\). Here is a simple C# program to demonstrate this solution: ```csharp using System; class Program { static void Main() { // Coefficients from the equations double a1 = 1.0; // coefficient of x in first equation double b1 = 1.0; // coefficient of y in first equation double c1 = 10.0; // constant term in first equation double a2 = 1.0; // coefficient of x in second equation double b2 = -1.0; // coefficient of y in second equation double c2 = 4.0; // constant term in second equation // Solving the system of equations double determinant = a1 * b2 - a2 * b1; if (determinant == 0) { Console.WriteLine("The system of equations has no unique solution."); } else { double x = (c1 * b2 - c2 * b1) / determinant; double y = (a1 * c2 - a2 * c1) / determinant; Console.WriteLine($"x = {x}"); Console.WriteLine($"y = {y}"); } } } ``` ### Explanation: 1. **Coefficients Extraction**: The coefficients from the equations are extracted and stored in variables. 2. **Determinant Calculation**: The determinant of the coefficient matrix is calculated to check if the system has a unique solution. 3. **Solution Calculation**: If the determinant is non-zero, the values of \(x\) and \(y\) are calculated using Cramer's rule. 4. **Output**: The results are printed to the console. This program demonstrates how to solve a system of linear equations using basic algebraic methods in C#.