CRAMER'S RULE FOR A 2x2 LINEAR SYSTEM
Consider the following system of linear equation.
a1x + b1y = c1
a2x + b2y = c2
Write a determinant Δ with the coefficients of x and y as shown below.
To get the determinant Δx, replace the first column elements a1, a2 of Δ with c1, c2 respectively and keep the second column elements b1, b2 of Δ as it is.
To get the determinant Δy, replace the second column elements b1, b2 of Δ with c1, c2 respectively and keep the first column elements a1, a2 of Δ as it is.
By Cramer's Rule,
Example :
Solve the following system of linear equations using Cramer's rule.
x + 2y = 3
x + y = 2
Solution :
Find the value of Δ :
Δ = 1 - 2
Δ = -1
Find the value of Δx :
Δx = 3 - 4
Δx = -1
Find the value of Δy :
Δy = 2 - 3
Δy = -1
By Cramer's Rule,
x = 1, y = 1
Note :
Cramer’s rule is applicable when Δ ≠ 0.
If Δ = 0, then the given system may be consistent or inconsistent.
Case 1 :
If Δ = 0 and Δx = 0, Δy = 0 and atleast one of the coefficients a11, a12, a21, a22 is non-zero, then the system is consistent and has infinitely many solutions.
Case 2 :
If Δ = 0 and atleast one of the values Δx, Δy is non-zero, then the system is inconsistent i.e. it has no solution.
Solve the following systems using Cramer's Rule.
Problem 1 :
3x + 2y = 5
x + 3y = 4
Solution :
Find the value of Δ :
Δ = 9 - 2
Δ = 7
Find the value of Δx :
Δx = 15 - 8
Δx = 7
Find the value of Δy :
Δy = 12 - 5
Δy = 7
By Cramer's Rule,
x = 1, y = 1
Problem 2 :
x + 2y = 3
2x + 4y = 8
Solution :
Find the value of Δ :
Δ = 4 - 4
Δ = 0
Find the value of Δx :
Δx = 12 - 16
Δx = -4
Since Δ = 0 and Δx ≠ 0, the system is consistent and it has no solution.
Problem 3 :
x + 2y = 3
2x + 4y = 6
Solution :
Find the value of Δ :
Δ = 4 - 4
Δ = 0
Find the value of Δx :
Δx = 12 - 12
Δx = 0
Find the value of Δy :
Δy = 6 - 6
Δy = 0
Since ∆ = 0, ∆x = 0, ∆y = 0 and ∆ has atleast one non-zero element, the system is consistent and it has infinitely many solutions.
Problem 4 :
2x + 4y = 6
6x + 12y = 24
Solution :
Find the value of Δ :
Δ = 24 - 24
Δ = 0
Find the value of Δx :
Δx = 72 - 96
Δx = -24
Since Δ = 0 and Δx ≠ 0, the system is consistent and it has no solution.
Problem 5 :
2x + y = 3
6x + 3y = 9
Solution :
Find the value of Δ :
Δ = 6 - 6
Δ = 0
Find the value of Δx :
Δx = 9 - 9
Δx = 0
Find the value of Δy :
Δy = 18 - 18
Δy = 0
Since ∆ = 0, ∆x = 0, ∆y = 0 and ∆ has atleast one non-zero element, the system is consistent and it has infinitely many solutions.
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