USING CRAMERS RULE TO SOLVE A SYSTEM OF EQUATIONS
Cramer's rule is one of the methods to solve system of equations. The values of the variables are found with the help of determinants.
The given system of equations will be in the form of AX = B.
A = the coefficient matrix
X = column matrix with unknown variables
B = Column matrix with constants.
Δ, Δx, Δy
- Δ ≠ 0, Δx ≠ 0, Δy ≠ 0 then the system will have unique solution.
- Δ = 0, Δx = 0, Δy = 0 then the system will have infinitely many solutions.
- Δ = 0, Δx or Δy ≠ 0 (or) Δ ≠ 0, Δx or Δy = 0 then the system is inconsistent and it has no solution.
Use Cramer's Rule to solve each system.
Problem 1 :
x - 5y = -5
-4x - 2y = 20
Solution:
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By Cramer's rule, we get
Hence the solution is (-5, 0).
Problem 2 :
-x + 5y = 2
x - 2y = -2
Solution:
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By Cramer's rule, we get
Hence the solution is (-2, 0).
Problem 3 :
2x + 2y = 0
4x - y = -20
Solution:
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By Cramer's rule, we get
Hence the solution is (-4, 4).
Problem 4 :
3x - 4y = 1
-5x + 2y = 3
Solution:
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By Cramer's rule, we get
Hence the solution is (-1, 1).
Problem 5 :
-x - y = -1
3x + 3y = 3
Solution:
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Since Δ = 0, Δx = 0 and Δy = 0 and atleast one of the element in Δ is non zero.
Then the system is consistent and it has infinitely many solution.
Problem 6 :
-5x + 5y = 10
-2x + 2y = -4
Solution:
Here, Δ = 0 but Δx ≠ 0.
So, the system is inconsistent and it has no solution.
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