With two unknowns you used elimination: scale the equations so a variable cancels, then add. Three unknowns need three equations, and the same idea works in two rounds.
Key idea
Pick one variable. Use two different pairs of equations to eliminate that same variable twice. That leaves two equations in the other two variables, which you solve as before. Then substitute back to find the last variable, and check all three equations.
Why it works
Adding a multiple of one true equation to another gives a new true equation, so elimination never changes the solutions. Each round removes one unknown. Picture each equation as a flat plane in space: the solution is the single point where all three planes meet.
| Step | What you do | What is left |
|---|---|---|
| 1 | (1) and (2): eliminate $x$ | equation (4) in $y$ and $z$ |
| 2 | (1) and (3): eliminate $x$ again | equation (5) in $y$ and $z$ |
| 3 | (4) and (5): eliminate $y$ | one equation in $z$ |
| 4 | back-substitute | $y$, then $x$ |
Worked examples
Worked example: A full solution
Solve $x + y + z = 6$, $2x - y + z = 3$, $x + 2y - z = 2$.
- (1) + (2) eliminates $y$: $3x + 2z = 9$. Call it (4).
- $2 \times$(1) $-$ (3) eliminates $y$ again: $2x + 2y + 2z - x - 2y + z = 12 - 2$, so $x + 3z = 10$. Call it (5).
- From (5), $x = 10 - 3z$. Into (4): $30 - 9z + 2z = 9$, so $z = 3$.
- Then $x = 1$, and from (1) $y = 6 - 1 - 3 = 2$.
- Check (2): $2 - 2 + 3 = 3$. Check (3): $1 + 4 - 3 = 2$.
Answer: $(1, 2, 3)$
Worked example: One equation is missing a variable
Solve $x + y = 5$, $y + z = 7$, $x + z = 8$.
- Add all three: $2x + 2y + 2z = 20$, so $x + y + z = 10$.
- Subtract each original: $z = 10 - 5 = 5$, $x = 10 - 7 = 3$, $y = 10 - 8 = 2$.
Answer: $(3, 2, 5)$
Common mistake
Eliminating $x$ from one pair and $y$ from another. Then both new equations still contain different leftover variables, and you are no closer. Decide which variable goes first and eliminate that same one from two different pairs. Check your final triple in all three original equations, not just the last one you used.