A point, a line, or nothing in common: Three planes can meet in exactly one point (unique solution), share no common point (no solution), or meet along a whole line (infinitely many). Elimination tells you which case you're in.
What elimination shows
- Unique → you get clean values for x, y, z (planes meet at a point)
- No solution → a false line like 0 = 5 (inconsistent)
- Infinitely many → a vanishing line 0 = 0 (a redundant equation)
IB-style question — infinitely many
Show that this system has infinitely many solutions:
x + y + z = 6, x + 2y + 3z = 14, 2x + 3y + 4z = 20.
Step by step
- Equation 2 − equation 1.
- Equation 3 − 2×(equation 1).
- These are the SAME equation — subtracting gives 0 = 0. One equation is redundant.
Final answer
Only two independent equations remain, so there are infinitely many solutions (the planes meet in a line).
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A false line means no solution: If elimination ever produces an impossible statement like 0 = 5, the equations contradict each other — there is no solution. (0 = 0 instead would mean infinitely many.)
IB-style question — no solution
Show that this system has no solution:
x + y + z = 6, x + 2y + 3z = 14, 2x + 3y + 4z = 25.
Step by step
- Equation 2 − equation 1.
- Equation 3 − 2×(equation 1).
- Subtract these two — the variables vanish but the numbers don't agree.
Final answer
0 = 5 is impossible, so the system is inconsistent — no solution.