Systems with Three Variables
Reduce to two variables then solve in sequence.
Key idea
Reduce to two variables then solve in sequence.
In depth
Reduce three equations to two by eliminating the same variable twice, solve the two-variable system, then back-substitute. Geometrically it is the intersection of three planes.
Worked example
Eliminate z from pairs of equations, solve for x and y, then find z.
Real-world uses
- Solving three-ingredient recipe or three-product sales puzzles
- Circuit analysis with several loops
Common mistake
Eliminating different variables in each step, which does not reduce the system.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Systems with Three Variables?
Reduce to two variables then solve in sequence. It is a topic at the advanced level in Algebra.
Where is Systems with Three Variables used in real life?
Real-world examples: Solving three-ingredient recipe or three-product sales puzzles; Circuit analysis with several loops.
What should I learn before Systems with Three Variables?
Make sure you are comfortable with Systems: Elimination, the previous topic in Algebra.
What is a common mistake with Systems with Three Variables?
Eliminating different variables in each step, which does not reduce the system.
How can I practise Systems with Three Variables?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Systems with Three Variables?
Move on to Systems of Inequalities next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.
- Quadratic Formula – Gives roots of any quadratic.