Chickens and rabbits: 8 animals in all, 22 legs in all — how many rabbits?
One condition leaves countless answers — the moment two conditions overlap, the answer squeezes to a single point.
Experiment
Hands-on experiment
🔮 Predict first — chickens and rabbits, 8 animals total, 22 legs total. How many rabbits?
🐔🐰 Vary the rabbits and count the legs
Heads stay at 8 (chickens + rabbits). Find the combo giving 22 legs. (Chickens: 2 legs, rabbits: 4.)
16
total legs · 0×4 + 8×2
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Problems with two unknowns are everywhere — chickens and rabbits, adult and child tickets, two unit prices. One expression (x+y=8) only narrows candidates; it doesn't decide.
The breakthrough: find one more condition. Add the legs condition (2x+4y=22) and exactly one of the nine candidates survives — the answer is the intersection of two conditions.
Then came elimination, the mechanical solver: add or subtract equations to erase one unknown and the problem collapses into one you know (a linear equation). This 'reduce the unknowns' strategy scales all the way to systems of hundreds (matrices).
Insight
Insights from the video
“One equation is a line; a system is an intersection.”
Graph x+y=8 and you get a line — every point on it a candidate. The second equation is another line, and their single crossing point is the system's solution. Algebra's elimination and geometry's intersection are one event.
“Elimination is the strategy of shrinking the unknown.”
Two unknowns are hard; one is easy. Subtract equations to erase one, and a hard problem falls into a known one — mathematics' signature move of converting the unknown into the known.
Misconception
Common misconceptions
Two unknowns can be solved from one equation.
x+y=8 alone admits (0,8), (1,7), … (8,0) — nine answers. You need as many independent conditions as unknowns for the answer to be pinned — the reason systems exist.
Systems are solved by trial and error.
Nine trials finish 8 animals — but 800? Elimination adds and subtracts equations to erase an unknown: a systematic method that ends in a few lines at any size. Trial discovers; elimination solves.
Formula
Writing it as math
What chickens and rabbits confirmed, in mathematical language.
Setting up the system
Chickens x, rabbits y. A heads condition and a legs condition — two unknowns demand two independent equations.
Elimination
Double the first equation and subtract: x vanishes → y=3 (rabbits), x=5 (chickens). Erase one unknown; land on a known problem.
The geometric meaning
Each equation is a line on the plane; the system's solution is their single intersection point.
In Real Life
Where you meet it in real life
Ticket receipts
Five adult+child tickets totaling $47, adults $12 and children $8 — how many of each is a classic system. One receipt hands you two equations.
Where two plans cross
The cost formulas of a basic and an unlimited plan meet at a point — the switch-over usage from the inequality lab was really a system's solution.
Mixing problems
Blend 6% and 10% saline into 400g of 8% — an amount condition plus a salt condition fixes the recipe. Cooking, chemistry, and pharmacy mixes are systems throughout.
GPS
Distance conditions from several satellites are solved together to pin your position to one point — more overlapping conditions, sharper location.
Try Yourself
Test yourself
Q1Two numbers sum to 20 and differ by 4. Find them.Show answer ▾
x+y=20, x−y=4. Add the equations: 2x=24 → x=12, y=8 — the cleanest example of eliminating by adding.
Q2Adult tickets $12, child $8. Five tickets cost $52 — how many adult tickets?Show answer ▾
x+y=5, 12x+8y=52. Subtract 8×(first): 4x=12 → 3 adults, 2 children.
Q3How many solutions does the system x+y=3, 2x+2y=6 have?Show answer ▾
Infinitely many — the second equation is twice the first: the same condition (the same line). Two equations that aren't independent don't squeeze the answer.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
Related video
Connection
Concepts connect
Previous concept
Leads to next
Linear Functions
Go watch x+y=8 become a line on the plane — the system's solution appears as a crossing point.
Go to the Linear Functions lab →Related
Labs worth exploring together
Related lab
Linear Functions
Solution = where two lines cross — one glance at the graph shows it.
Go to the Linear Functions lab →Related lab
The Coordinate Plane
The stage where equations become lines — algebra meets geometry.
Go to the The Coordinate Plane lab →