Systems of linear equations
A system of linear equations is a set of equations that must all be true at the same time, such as 2x + 3y = 13 and x − y = −1. Each linear equation in two unknowns is a straight line, and the solution is the point where the lines cross. With three unknowns, each equation is a plane, and the solution is the point shared by all three planes.
Cramer’s rule
The calculator uses Cramer’s rule, which expresses each unknown as a ratio of determinants:
- D is the determinant of the coefficients.
- Dₓ, Dᵧ and D_z are the same determinant with the column of x, y or z replaced by the right-hand sides.
- x = Dₓ ÷ D, y = Dᵧ ÷ D, z = D_z ÷ D.
The rule works as long as D is not zero. For larger systems, elimination is faster, but for two or three unknowns Cramer’s rule is quick and exact.
Worked example with two unknowns
Solve 2x + 3y = 13 and x − y = −1.
- D = 2 × (−1) − 3 × 1 = −5.
- Dₓ = 13 × (−1) − 3 × (−1) = −10, so x = −10 ÷ −5 = 2.
- Dᵧ = 2 × (−1) − 13 × 1 = −15, so y = −15 ÷ −5 = 3.
- Check: 2 × 2 + 3 × 3 = 13 and 2 − 3 = −1.
Worked example with three unknowns
Solve 2x + 3y − z = 13, x − y + 2z = −1 and x + y + z = 6. The determinant is −5, and the solution is x = 1, y = 4, z = 1. Substituting back: 2 + 12 − 1 = 13, 1 − 4 + 2 = −1 and 1 + 4 + 1 = 6.
When there is no unique solution
| Case | Determinant | Picture with two unknowns |
|---|---|---|
| Unique solution | not zero | two lines crossing at one point |
| No solution | zero | two parallel lines |
| Infinitely many solutions | zero | the same line written twice |
For example, x + y = 2 and 2x + 2y = 5 have no solution, because the second line is parallel to the first. The calculator flags a zero determinant instead of returning a misleading answer.
Entering your equations
Write each equation in the form ax + by + cz = d, moving every term with an unknown to the left and the constants to the right. A missing unknown has a coefficient of 0, and a term like −y has a coefficient of −1.
Where systems appear
Systems of equations solve mixture problems (how many liters of each solution to combine), pricing puzzles (two tickets and a drink cost this much), current in electric circuits with Kirchhoff’s laws, and balancing chemical equations.