Systems of three linear equations in three variables

Some problems involve three unknowns, like the numbers of three kinds of tickets sold, and translating each fact in the problem into an equation gives a system of three linear equations in three variables. Such a system can have exactly one solution, infinitely many, or none; geometrically, each equation is a plane, and the solution is where all three planes meet. To solve one, use elimination to remove the same variable from two different pairs of equations, which leaves two equations in two variables, then solve those and substitute back. Substitution also works, the same steps done with row operations on an augmented matrix are called Gaussian elimination, and technology can solve the system from its matrix. For example, for x + y + z = 6, x - y + z = 2, and 2x + y - z = 1: subtracting the second equation from the first gives 2y = 4, so y = 2; then the first becomes x + z = 4 and the third becomes 2x - z = -1, and adding those gives 3x = 3, so x = 1 and z = 3. Always check the solution, (1, 2, 3), in all three original equations.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report