Cramer's Rule Calculator

Cramer's Rule Calculator calculates the value of the variables for the given linear equations. A linear equation is defined as an equation that is written for two different variables. This equation will be a linear combination of these two variables and a constant.

What is Cramer's Rule Calculator?

Cramer's Rule Calculator is an online tool that helps to calculate the value of the variables for the given linear equations. This online Cramers rule calculator helps you to calculate the value of the variables in a few seconds. To use this Cramer's rule calculator, please enter the coefficients in the given input box.

How to Use Cramer's Rule Calculator?

Please follow the steps below to find the value of variables using an online Cramer's rule calculator:

• Step 1: Go to Cuemath’s online Cramer's rule calculator.
• Step 2: Enter the coefficients of equations in the given input box of Cramer's rule calculator.
• Step 3: Click on the "Solve" button to find the value of variables.
• Step 4: Click on the "Reset" button to clear the fields and enter new values.

How Cramers Rule Calculator Works?

Cramer's rule is used for solving linear equations and find the values of variables for given linear equations.

Let A₁x + B₁y = C₁ and A₂x + B₂y = C₂ be the linear equations.

The formula used to solve the variables for the given two linear equations using the Cramers rule is given by

x = ∆x/∆ and y = ∆y/∆

\(∆ =\left|\begin{array}{ll} A_{1} & B_{1} \\ A_{2} & B_{2} \end{array}\right| ,\,\,∆x=\left|\begin{array}{ll} C_{1} & B_{1} \\ C_{2} & B_{2} \end{array}\right| \,\,and \,\,∆y=\left|\begin{array}{ll} A_{1} & C_{1} \\ A_{2} & C_{2} \end{array}\right|\)

There are two conditions for the Cramers rule:

Condition 1: If all the determinants are zero, then the system is consistent and has infinitely many solutions.

Condition 2: If ∆=0 and ∆x & ∆y are not equal to zero, then the system is inconsistent and the equations do not have any solution.

Let us understand this with the help of the following example.

Solved Example on Cramer's rule

Solve the given linear equations x − 2y = -3  and 3x − 4y = -5  using Cramer's rule and verify it using the Cramer's rule calculator?

Solution:

Given: A₁ = 1, B₁ = -2, C₁ = -3, A₂ = 3, B₂ = -4, C₂ = -5

x = ∆x/∆ and y = ∆y/∆

\(∆ =\left|\begin{array}{ll} A_{1} & B_{1} \\ A_{2} & B_{2} \end{array}\right| ,\,\,∆x=\left|\begin{array}{ll} C_{1} & B_{1} \\ C_{2} & B_{2} \end{array}\right| \,\,and \,\,∆y=\left|\begin{array}{ll} A_{1} & C_{1} \\ A_{2} & C_{2} \end{array}\right|\)

\(∆ =\left|\begin{array}{ll} 1 & -2 \\ 3 & -4 \end{array}\right| ,\,\,∆x=\left|\begin{array}{ll} -3 & -2 \\ -5& -4 \end{array}\right| \,\,and \,\,∆y=\left|\begin{array}{ll} 1& -3 \\ 3 & -5 \end{array}\right|\)

∆ = 2, ∆x = 2, ∆y = 4

x = ∆x/∆ = 2/2 = 1

y = ∆y/∆ = 4/2 = 2

Therefore, the value of x and y are (1,2)

Now, try the cramers rule calculator and find the value of the variables for:

• 2x + 5y = 6 and 4x  - 5y = 10
• -4x - 10y = 7 and 5x + 5y = 9

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