Systems of Three Equations
Solve three unknowns by eliminating variables step by step.
About this topic
A system of three linear equations in x, y and z represents three planes in space. Solve by using elimination to remove one variable from pairs of equations, leaving a two-variable system, then solve that and back-substitute.
The planes may meet at a single point, along a line (infinitely many solutions), or not at all. Systems like these appear when balancing three constraints, like mixing three ingredients, and are efficiently solved with matrices.
Worked example: solve x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2. Add the first and third equations to eliminate z: 2x + 3y = 8. Add the second and third to eliminate z again: 3x + y = 5. Now solve the pair: y = 5 − 3x, so 2x + 15 − 9x = 8, giving x = 1, y = 2, and z = 3. Check in all three equations before finishing.
Engineers and economists solve such systems for currents in circuits, mixtures and budgets. Keep the same variable order in each equation and carry minus signs carefully, since most mistakes are arithmetic. If elimination gives something false like 0 = 5, the planes have no common point; if it gives 0 = 0, there are infinitely many solutions. Do not stop after finding two variables, because the third must also be found and checked.
Key ideas
- Eliminate one variable to get two equations
- Solve the 2 × 2 system
- Back-substitute
Quick check
Use cases: where Systems of Three Equations helps
- Homework: review the key idea behind Systems of Three Equations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Systems of Three Equations without notes.
- Visual learning: play with the interactive lab to see how Systems of Three Equations changes when you adjust its inputs.
- Teaching a friend: explain Systems of Three Equations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Systems of Three Equations?
Solve three unknowns by eliminating variables step by step.
What will I learn in Systems of Three Equations?
Eliminate one variable to get two equations. Solve the 2 × 2 system. Back-substitute.
Is Systems of Three Equations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Systems of Three Equations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Systems of Three Equations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.