Systems of three linear equations appear in Class 12 matrices, engineering mathematics and circuit analysis. Enter the four numbers of each equation to solve for x, y and z.
Write each equation as ax + by + cz = d.
Enter a, b, c, d for each equation separated by commas.
Read x, y and z.
Cramer's rule: x = Dx ÷ D, y = Dy ÷ D, z = Dz ÷ D
This solver handles three linear equations in three unknowns, x, y and z, each written as ax + by + cz = d. You enter the four numbers for each equation, and it computes the coefficient determinant D and the three replaced determinants Dx, Dy and Dz, then applies Cramer's rule: x = Dx ÷ D, y = Dy ÷ D and z = Dz ÷ D.
Three-variable systems are a standard Class 12 matrices and determinants exercise and appear in engineering mathematics, electrical circuit analysis using Kirchhoff's laws, chemistry balancing, and cost problems with three products. Solving them by hand takes many steps and one sign error spoils everything, so the calculator is useful for checking homework and for seeing whether a system has a unique solution at all.
1. Write each equation in the form ax + by + cz = d, using 0 for any missing variable.
2. Form the 3 × 3 coefficient matrix from the a, b and c values.
3. Compute D by expanding along the first row: D = a₁(b₂c₃ − b₃c₂) − b₁(a₂c₃ − a₃c₂) + c₁(a₂b₃ − a₃b₂).
4. If D = 0, the system has no unique solution.
5. Form Dx, Dy and Dz by replacing the x, y or z column with the constants d₁, d₂, d₃, and evaluate each determinant.
6. Compute x = Dx ÷ D, y = Dy ÷ D and z = Dz ÷ D.
7. Substitute all three values into each equation to check.
Write the system as AX = B. The determinant is linear in each column. If you replace the first column of A by B, that column equals x times column 1 plus y times column 2 plus z times column 3 of A. The y and z parts each give a determinant with two equal columns, which is zero, so Dx = x × D. Therefore x = Dx ÷ D, and the same reasoning gives y and z. This is Cramer's rule for any size of system.
NCERT Class 12 also solves AX = B by computing X = A⁻¹B, where A⁻¹ = adj(A) ÷ |A|. Writing that product out term by term reproduces the Cramer fractions exactly, because each entry of adj(A)B is a determinant with one column replaced. So the two methods are the same calculation arranged differently. Both need |A| ≠ 0. Gaussian elimination, reducing the system row by row, is faster for large systems and is what computers usually use.
Each equation describes a plane in three-dimensional space. A unique solution means the three planes meet at a single point. If D = 0, the planes do not meet in exactly one point: they may share a whole line, coincide, or have no common point at all, for example when two are parallel. To tell these cases apart, check whether Dx, Dy and Dz are also zero and look for a contradiction by elimination. The calculator reports only that no unique solution exists.
Aditya's Class 12 practice sheet asks him to solve x + y + z = 12, 2x − y + 3z = 17 and 3x + 2y − z = 12 by Cramer's rule.
Coefficient determinant D: D = 13
Replace each column with the constants: Dx = 39, Dy = 52, Dz = 65
Cramer's rule: x = 39 ÷ 13 = 3 y = 52 ÷ 13 = 4 z = 65 ÷ 13 = 5
Answer: Solution x = 3, y = 4, z = 5; Determinant D 13
Leaving out a zero for a missing variable, which shifts every later number into the wrong column.
Forgetting the minus sign on the middle term when expanding a 3 × 3 determinant.
Entering the constant on the wrong side, such as from ax + by + cz − d = 0, without changing its sign.
Replacing a row instead of a column when forming Dx, Dy or Dz by hand.
Assuming D = 0 always means no solution. It can also mean infinitely many.
Checking Class 12 determinant and matrix-method answers.
Solving for three loop currents in circuits using Kirchhoff's laws.
Finding prices of three items from three different bills.
Fitting a parabola y = ax² + bx + c through three given points.
Balancing mixtures or diets with three ingredients to meet three targets.
What if a variable is missing from an equation?
Enter 0 for its coefficient.
Is this the same as using a matrix inverse?
Yes, both methods give identical answers for a non-singular system.