The Cramer’s Rule Calculator (2 x 2) is an online tool that finds the solution of linear equations in two variables, by finding the determinant of the coefficient matrix. We need to enter the real coefficients of the equations, in the input field to get the output. CoolGyan’S free online Calculator makes it easy to find the solutions of equations, simply and easily.
Cramer’s Rule Formula: If we are given with two linear equations,
a1x + b1y = c1
a2x + b2y = c2
Then the main determinant of the 2×2 matrix formed by the coefficients of linear equations is given by:
\(∆ =\left|\begin{array}{ll}
a_{1} & b_{1} \\
a_{2} & b_{2}
\end{array}\right|\)
The other two determinants are:
\(∆x=\left|\begin{array}{ll}
c_{1} & b_{1} \\
c_{2} & b_{2}
\end{array}\right| \quad \text { and } \quad ∆y=\left|\begin{array}{ll}
a_{1} & c_{1} \\
a_{2} & c_{2}
\end{array}\right|\)
Therefore, the solution of the two given equations are:
X = ∆x/∆ and Y = ∆y/∆
If all the determinants are zero, then the equations are dependent and the system is consistent and have infinitely many solutions. If ∆=0 and ∆x & ∆y are not equal to zero, then the system is inconsistent and the equations do not have any solution.
How to Solve Linear Equations Using Determinants?
To find the solution of linear equations using the determinant of coefficients, with the help of calculators, follow the below steps.
Step 1: Write all the coefficients of linear equations in the respective input fields.
Step 2: Click on “Solve these equations” button
Step 3: The value of the main determinant and x & y determinants will appear in the respective fields. Also, the solution of equations, i.e. value of x and y will appear in the respective output field.
Solved Example
Find the solution of the given equations, using Cramer’s rule:
2x+3y = 4
x+2y = 3
Solution: a1 = 2, b1 = 3 and c1 = 4
a2 = 1, b2 = 2 and c2 = 3
Determinant,
∆ = 1
The other two determinants are:
∆x=-1
∆y=2
Therefore, the solution of the two given equations are:
x = ∆x/∆ = -1/1 = -1
and y = ∆y/∆ = 2/1 = 1
