{ } · Simultaneous Equations

Where two conditions overlap, a single point — simultaneous equations.

Chickens and rabbits: 8 animals in all, 22 legs in all — how many rabbits?

Find a solution shared by two equations.

The chicken-and-rabbit puzzle from a Chinese math classic 1,500 years old.

A reading path containing this lesson: Unknown numbers and balance · 3/3

Why does this question matter?

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).

Pause and try it yourself

Make a prediction first. Change one input at a time and compare what happens with what you expected.

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.)

rabbits08 chickens

🐔🐔🐔🐔🐔🐔🐔🐔

total legs · 0×4 + 8×2

Read the idea from another angle

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.

Now read the definitions with their conditions

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.

Make one more distinction

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.

Where else can this idea help?

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.

Explain it in your own words

Try another question

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.