System of 3 Equations Solver

Table of Contents

• System of 3 equations definition
• Parts of a system of 3 equations
  • Coefficients and constants
  • Determinant (D)
  • Determinants Dx, Dy and Dz
  • Solution (x, y, z)
• Solving with Cramer's rule
• Number of solutions
• Frequently asked questions

System of 3 equations definition

A system of 3 equations is a set of three linear equations with the same three unknowns, x, y and z, that must all be true at once. In standard form:

a1 * x + b1 * y + c1 * z = d1

a2 * x + b2 * y + c2 * z = d2

a3 * x + b3 * y + c3 * z = d3

Each equation describes a flat plane in space, and the solution is the single point where all three planes meet. Such systems appear in physics (the currents in a circuit), chemistry (balancing reactions), economics and computer graphics.

For fewer unknowns, see the linear equation and the system of 2 equations solvers.

Parts of a system of 3 equations

You enter the twelve numbers of the three equations, and the tool shows x, y and z. On the way, it calculates four determinants in the background: D, Dx, Dy and Dz.

Coefficients and constants

The coefficients are the numbers that multiply the unknowns: a1, a2 and a3 go with x, b1, b2 and b3 with y, and c1, c2 and c3 with z. The number says which equation they belong to, and a missing term has a coefficient of 0.

The constants d1, d2 and d3 are the numbers on the right side of the equations.

Determinant (D)

The determinant (D) is a single number calculated from the nine coefficients. It decides whether the system has exactly one solution: it does only when D is not zero. When D = 0, this tool shows no solution.

D = (a1 * b2 * c3) + (b1 * c2 * a3) + (c1 * a2 * b3) - (c1 * b2 * a3) - (a1 * c2 * b3) - (b1 * a2 * c3)

This is the rule of Sarrus: the three products running diagonally down to the right are added, and the three running down to the left are subtracted.

Determinants Dx, Dy and Dz

Dx, Dy and Dz are calculated the same way as D, with one column of coefficients replaced by the constants: the x column (a1, a2, a3) for Dx, the y column for Dy, and the z column for Dz. For example:

Dx = (d1 * b2 * c3) + (b1 * c2 * d3) + (c1 * d2 * b3) - (c1 * b2 * d3) - (d1 * c2 * b3) - (b1 * d2 * c3)

Solution (x, y, z)

The solution is the set of values that makes all three equations true. Each unknown is its own determinant divided by D:

x = Dx / D, y = Dy / D, z = Dz / D

Solving with Cramer's rule

This tool uses Cramer's rule, which solves the system with determinants alone. Take this example:

x + y + z = 6

2 * x - y + z = 3

x + 2 * y - z = 2

1. The determinant D. The three products running down to the right are 1, 1 and 4, and the three running down to the left are -1, 2 and -2:

   D = (1 + 1 + 4) - (-1 + 2 - 2) = 7

   D is not zero, so there is exactly one solution.

2. Dx, Dy and Dz. Put the constants 6, 3 and 2 in place of each column in turn:

   Dx = 7, Dy = 14, Dz = 21

3. Divide by D.

   x = 7 / 7 = 1

   y = 14 / 7 = 2

   z = 21 / 7 = 3

Putting the answer back into the three equations confirms it: 1 + 2 + 3 = 6, 2 - 2 + 3 = 3 and 1 + 4 - 3 = 2.

By hand, the same system can also be solved by elimination: remove the same unknown from two pairs of equations, which leaves a system of 2 equations, then solve that and work back.

Number of solutions

Three planes can meet in three ways:

• One solution: they meet in a single point, and D is not zero.
• No solution: at least two planes are parallel, or they meet in pairs along three parallel lines, so no point lies on all three. D = 0.
• Infinitely many solutions: the three planes share a whole line, or are the same plane. D = 0.

Frequently asked questions

How do you solve a system of three equations?

With Cramer's rule: calculate the determinant D of the coefficients and the three determinants Dx, Dy and Dz, then divide, x = Dx / D, y = Dy / D and z = Dz / D. By hand, elimination reduces it to a system of two equations first.

What is Cramer's rule?

A method that solves a system of linear equations with determinants. Each unknown is the determinant with that unknown's column replaced by the constants, divided by the determinant of the coefficients. It works only when that determinant is not zero.

How do you calculate a 3 × 3 determinant?

Use the rule of Sarrus: add the products of the three diagonals running down to the right, and subtract the products of the three running down to the left. The full formula is in the Determinant (D) section above.