What is a system in three variables?
A system of linear equations in three variables is a set of three equations, each of the form ax + by + cz = d, that must be true at the same time. Geometrically each equation is a flat plane in space, and a unique solution is the single point (x, y, z) where all three planes intersect.
Key formula
General equation: ax + by + cz = d. A consistent, independent system has exactly one solution (x, y, z).
The elimination method
The reliable strategy is to remove one variable at a time:
- Pair up two equations and eliminate one variable (say z) to get an equation in x and y.
- Pair a different two equations and eliminate the same variable again.
- You now have two equations in two variables — solve them, then substitute back to find the third variable.
Worked example
Solve: x + y + z = 6, x + 2y + 3z = 14, x + 4y + 9z = 36.
Subtract eq1 from eq2: y + 2z = 8. Subtract eq2 from eq3: 2y + 6z = 22, i.e. y + 3z = 11. Subtract these: z = 3. Then y = 8 − 2(3) = 2, and x = 6 − 2 − 3 = 1.
Solution: x = 1, y = 2, z = 3. Check: 1 + 8 + 27 = 36. ✓
Substitution and checking
If one equation already gives a variable (for example x = 2), substitute it into the other two to reduce the system quickly. Always substitute your final answer back into all three original equations — a correct triple must satisfy every one of them.
Remember
- Eliminate the same variable both times, or you will not reduce the system.
- A unique solution means the three planes meet at one point.
- Keep equations lined up by variable to avoid arithmetic slips.