METHODS FOR SOLVING TWO EQUATIONS METHODS FOR SOLVING THREE EQUATIONS EXAMPLES

1. APPLIED MATHEMATICS IN CHEMICAL
ENGINEERING (HCE21)
SEMINAR ON:
CRAMER’S RULE
DONE BY
KISHAN KASUNDRA
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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,
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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
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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
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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
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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)
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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
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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
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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).
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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
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11. THANK
YOU!!
YOU MAY CLAP YOUR HANDS NOW
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