1. F UNDAMENTALS OF E NGINEERING A NALYSIS Eng. Hassan S. Migdadi Cramer’s Rule for solving linear systems Part 1

2. Coefficient Matrices You can use determinants to solve a system of linear equations. You use the coefficient matrix of the linear system. Linear SystemCoeff Matrix ax+by=e cx+dy=f

3. Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear SystemCoeff Matrix ax+by=e cx+dy=f If detA0, then the system has exactly one solution: and

4. Example 1- Cramer’s Rule 2x2 Solve the system: 8x+5y=2 2x-4y=-10 The coefficient matrix is: and So: and

5. Solution: (-1,2)

6. Example 2- Cramer’s Rule 2x2 Solve the system: 2x+y=1 3x-2y=-23 The solution is: (-3,7) !!!

7. Example 3- Cramer’s Rule 3x3 Solve the system: x+3y-z=1 -2x-6y+z=-3 3x+5y-2z=4 Let’s solve for Z Z=1 The answer is: (-2,0,1)!!!

8. Solving 3D Linear System using Cramer ’ s Rule

9. Example: 2x + y + z = 3 x – y – z = 0 x + 2y + z = 0

10. Determinants of XYZ =3 =-6 =9 X= 1, y =-2, z=3

11. Given the following system of equations, find the value of z. 2x + y + z = 1 x – y + 4z = 0 x + 2y – 2z = 3 =-3=-6 Z=2