Kishan Kasundra presented on Cramer's rule for solving systems of equations. Cramer's rule uses determinants of coefficient matrices to find variables. It can be applied to systems with 2 or 3 equations. For 2 equations, the determinant of the original coefficient matrix and matrices with the coefficient columns replaced by the constants are calculated. The ratios of these determinants over the original give the variables. For 3 equations, a similar process is followed using a 3x3 coefficient matrix. An example showed applying Cramer's rule to solve a 2x2 system.

1. APPLIED MATHEMATICS IN CHEMICAL
ENGINEERING (HCE21)
SEMINAR ON:
CRAMER’S RULE
DONE BY
KISHAN KASUNDRA
4/21/2017 1CHEMICAL ENGINEERING DEPARTMENT, DSCE

2. METHOD FOR SOLVING TWO EQUATIONS:-
To find out variables, we need to have equations which are
useful to solve the values of variables.
First we assume two equations,
a1x +b1y = c1
a2x + b2y = c2
Then we need to find determinant ‘D’ of coefficient matrix, for
that construct 2x2 matrix
Now, for determinant first we need to solve this matrix by cross
multiplication,
4/21/2017 2CHEMICAL ENGINEERING DEPARTMENT, DSCE

3. Now, replace the x-coefficient column with the constant
column in coefficient matrix.
Now, replace the y-coefficient column with the constant
column in coefficient matrix.
Now, according to Cramer’s rule,
x= 𝑑𝑒𝑡𝐷𝑥/𝑑𝑒𝑡𝐷 & y = 𝑑𝑒𝑡𝐷𝑦/𝑑𝑒𝑡𝐷
4/21/2017 3CHEMICAL ENGINEERING DEPARTMENT, DSCE

4. METHOD FOR SOLVING THREE EQUATIONS:-
To find out variables, we need to have equations which are
useful to solve the values of variables.
First we assume two equations,
a1x +b1y + c1z = d1
a2x + b2y +c2z = d2
a3x + b3y + c3z = d3
First, we need to find determinant ‘D’ of coefficient matrix,
for that construct 3x3 matrix
4/21/2017 4CHEMICAL ENGINEERING DEPARTMENT, DSCE

5. Now, for determinant first we need to solve this matrix,
det D= a1(b2c3 – b3c2) – b1(a2c3 – c2a3) + c1 (a2b3 – b2a3)
Now, replace the x-coefficient column with the constant
column in coefficient matrix
4/21/2017 5CHEMICAL ENGINEERING DEPARTMENT, DSCE

6. Now, replace the y-coefficient column with the constant
column in coefficient matrix.
det Dy = a1(d2c3 – d3c2) – d1(a2c3 – c2a3) + c1 (a2d3 – d2a3)
4/21/2017 6CHEMICAL ENGINEERING DEPARTMENT, DSCE

7. Now, replace the z-coefficient column with the constant column
in coefficient matrix.
det Dz = a1(b2c3 – b3c2) – b1(a2c3 – c2a3) + c1 (a2b3 – b2a3)
Now, according to Cramer’s rule,
x= 𝑑𝑒𝑡𝐷𝑥/𝑑𝑒𝑡𝐷 y = 𝑑𝑒𝑡𝐷𝑦/𝑑𝑒𝑡𝐷 z = 𝑑𝑒𝑡𝐷z/𝑑𝑒𝑡𝐷
4/21/2017 7CHEMICAL ENGINEERING DEPARTMENT, DSCE

8. EXAMPLES:-
1) Solve the system by using Cramer’s rule:
3x – 2y = 4
2x + y = -3
SOLUTION:-
First find the determinants D, Dx, and Dy :
D=(3)(1) – (-2)(2) = 7
4/21/2017 8CHEMICAL ENGINEERING DEPARTMENT, DSCE

9. Dx= (4)(1) – (-2)(-3) = -2
Dy = (3)(-3) – (4)(2) = -17
By Cramer’s rule, we have
x = 𝐷𝑥/𝐷 = −2/7
y = 𝐷𝑦/𝐷 = −17/7
The solution is (x,y) = (-2/7, -17/7).
4/21/2017 9CHEMICAL ENGINEERING DEPARTMENT, DSCE

10. REFERENCES:-
http://www.coolmath.com/algebra/14-determinants-
cramers-rule/01-determinants-cramers-rule-2x2-05
http://www.sparknotes.com/math/algebra2/systemsofthreee
quations/section3.rhtml
http://www.wikihow.com/Find-the-Determinant-of-a-3X3-
Matrix
https://en.wikipedia.org/wiki/Cramer%27s_rule
http://www.purplemath.com/modules/cramers.htm
http://www.mhhe.com/math/devmath/dugopolski/acs/stude
nt/olc/graphics/dugopolski02acs_s/ch04/others/ch04-5.pdf
4/21/2017 10CHEMICAL ENGINEERING DEPARTMENT, DSCE

11. THANK
YOU!!
YOU MAY CLAP YOUR HANDS NOW
4/21/2017 11CHEMICAL ENGINEERING DEPARTMENT, DSCE