1. Fundamentals of Engineering Analysis
Cramer’s Rule for solving linear systems
Part 1
Eng. Hassan S. Migdadi

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

3. Cramer’s Rule for 2x2 System
Let A be the coefficient matrix
Linear System Coeff Matrix
ax+by=e
cx+dy=f
If detA 0, 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 =3
X= 1, y =-2, z=3
=-6 =9

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