Free Cramer's Rule Calculator

Solving Linear Equations with Cramer's Rule

Systems of linear equations appear frequently in mathematics, engineering, and everyday problem solving. This online system of equations solver applies Cramer's rule—a technique that works for both 2×2 and 3×3 cases and relies purely on arithmetic (addition, subtraction, multiplication, and division). No substitution, elimination, or graphing is involved. Whether you need a Cramers rule 2x2 or a Cramers rule 3x3 solution, the calculator provides the answer quickly by using matrix determinants.

What Is a System of Equations?

In algebra, an equation relates an unknown quantity to known numbers. When there are multiple unknowns and multiple relationships among them, we write several equations that must be satisfied simultaneously. For instance, if you know that two different purchases lead to different totals, you can set up equations such as:

$$2x + 3y = 7 \quad\text{and}\quad x - 4y = -2.$$

Here $x$ and $y$ represent unknown prices, and solving the system means finding the pair $(x,y)$ that makes both equalities true. The simplest systems have one equation per variable; the method of Cramer's rule handles exactly such situations when the equations are linear.

Using Matrices to Represent a System

A linear system can be compactly represented with matrices. The coefficients of the variables fill a matrix called the coefficient matrix. For a system with three equations and three unknowns, the coefficient matrix $W$ has three rows and three columns. Adding the constants from the right sides yields the augmented coefficient matrix, which is often displayed with a vertical separator.

Consider a travel-themed example where $x$ (bikes), $y$ (cars), and $z$ (planes) represent unknown counts:

\begin{cases} x + y + z = 26,\$$2pt] y - 2z = 6,\$$2pt] 2x - z = 12. \end{cases}

The coefficient matrix is

$$W = \begin{pmatrix} 1 & 1 & 1\\ 0 & 1 & -2\\ 2 & 0 & -1 \end{pmatrix},$$

and the augmented matrix is

$$\left(\begin{array}{ccc|c} 1 & 1 & 1 & 26\\ 0 & 1 & -2 & 6\\ 2 & 0 & -1 & 12 \end{array}\right).$$

Cramer's Rule Explained

Cramer's rule gives the solution of a square linear system as a ratio of determinants. For the system $W\mathbf{x} = \mathbf{d}$ (where $W$ is the coefficient matrix and $\mathbf{d}$ is the column of constants), the value of the variable $x_i$ is

$$x_i = \frac{\det(W_i)}{\det(W)},$$

where $W_i$ is the matrix obtained by replacing the $i$-th column of $W$ with $\mathbf{d}$.

Determinant of a 2×2 Matrix

For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$,

$$\det(A) = ad - bc.$$

Determinant of a 3×3 Matrix

For $B = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$,

$$\det(B) = a e i + b f g + c d h - c e g - b d i - a f h.$$

A handy way to remember this is Sarrus' rule: write the first two columns again to the right of the matrix, sum the three downward diagonals, and subtract the three upward diagonals.

Applying Cramer's Rule

To solve a system:

1. Compute the main determinant $\det(W)$.
2. For each variable, replace the corresponding column of $W$ with the constants to form a new matrix, and compute its determinant.
3. Divide the variable determinant by the main determinant.

If the main determinant equals zero, the system has either no solution or infinitely many solutions; this tool will report which situation holds, and in the 2×2 case with many solutions it can also show the form of the solution set.

Cramer's Rule for 2×2 Systems

A 2×2 system

$$\begin{cases} a_1 x + b_1 y = d_1,\\ a_2 x + b_2 y = d_2 \end{cases}$$

has coefficient matrix $\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}$. Then

$$x = \frac{ \det\begin{pmatrix} d_1 & b_1 \\ d_2 & b_2 \end{pmatrix} }{ \det\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} },\qquad y = \frac{ \det\begin{pmatrix} a_1 & d_1 \\ a_2 & d_2 \end{pmatrix} }{ \det\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} }.$$

Worked Example: 3×3 System

Now we apply Cramer's rule to the travel problem.

Step 1: Main determinant.

$$\begin{aligned} \det(W) &= 1\cdot1\cdot(-1) + 1\cdot(-2)\cdot2 + 1\cdot0\cdot0 \\ &\qquad - 1\cdot1\cdot2 - 1\cdot0\cdot(-1) - 1\cdot(-2)\cdot0 \\ &= -1 - 4 + 0 - 2 - 0 - 0 = -7. \end{aligned}$$

Step 2: Determinant for $x$ (replace column 1 with constants).

$$W_x = \begin{pmatrix} 26 & 1 & 1\\ 6 & 1 & -2\\ 12 & 0 & -1 \end{pmatrix},\qquad \begin{aligned} \det(W_x) &= 26\cdot1\cdot(-1) + 1\cdot(-2)\cdot12 + 1\cdot6\cdot0 \\ &\qquad - 1\cdot1\cdot26 - 1\cdot6\cdot(-1) - 26\cdot(-2)\cdot0 \\ &= -26 - 24 + 0 - 26 + 6 - 0 = -56. \end{aligned}$$

Step 3: Determinant for $y$ (replace column 2).

$$W_y = \begin{pmatrix} 1 & 26 & 1\\ 0 & 6 & -2\\ 2 & 12 & -1 \end{pmatrix},\qquad \begin{aligned} \det(W_y) &= 1\cdot6\cdot(-1) + 26\cdot(-2)\cdot2 + 1\cdot0\cdot12 \\ &\qquad - 1\cdot6\cdot2 - 26\cdot0\cdot(-1) - 1\cdot(-2)\cdot12 \\ &= -6 - 104 + 0 - 12 - 0 + 24 = -98. \end{aligned}$$

Step 4: Determinant for $z$ (replace column 3).

$$W_z = \begin{pmatrix} 1 & 1 & 26\\ 0 & 1 & 6\\ 2 & 0 & 12 \end{pmatrix},\qquad \begin{aligned} \det(W_z) &= 1\cdot1\cdot12 + 1\cdot6\cdot2 + 26\cdot0\cdot0 \\ &\qquad - 26\cdot1\cdot2 - 1\cdot0\cdot12 - 1\cdot6\cdot0 \\ &= 12 + 12 + 0 - 52 - 0 - 0 = -28. \end{aligned}$$

Step 5: Solution ratios.

$$x = \frac{-56}{-7} = 8,\qquad y = \frac{-98}{-7} = 14,\qquad z = \frac{-28}{-7} = 4.$$

Thus there are 8 bikes, 14 cars, and 4 planes. Substituting these back into the original equations confirms they satisfy all three equalities.

Using the Linear Equations Calculator

This tool is a convenient matrix determinant solver that handles both 2×2 and 3×3 systems. Simply enter the coefficients from your equations into the designated fields (the calculator accepts only linear equations—no quadratic terms or square roots of variables). It automatically computes the four determinants required by Cramer's rule and displays the solution. This makes it an ideal linear equations calculator for students, engineers, and anyone who needs to solve linear equations with matrices quickly.