Ch 1.4.4 — Of the Resolution of Two or More Equations of the First Degree
Summary: Euler develops systematic methods for solving simultaneous linear equations in two or more unknowns: the substitution method, explicit general formulas for two-variable systems, recursive elimination for three or more variables, and an auxiliary-variable trick (introducing the sum ) that greatly simplifies certain problems.
Sources: chapter-1.4.4
Last updated: 2026-05-03
Setup (§605–606)
When a problem introduces unknowns, we need exactly independent equations. All linear equations (no powers above 1, no products of unknowns) have the standard form
The chapter begins with two unknowns and then extends to three and beyond.
Substitution method for two unknowns (§607–608)
Given
express from each equation and equate:
Cross-multiplying and collecting gives the general solution:
These formulas are the 2×2 case of what is now called Cramer’s rule, derived here purely by algebraic manipulation without any matrix language. (source: chapter-1.4.4, §608)
Q1 (§609): Sum of two numbers = 15, difference = 7. , . From the first, ; from the second, . Equating: , .
Q2 (§610–611): The general version yields the theorem:
i.e., the greater is half the sum plus half the difference, the lesser is half the sum minus half the difference. Alternatively (§611), add the two equations to get , subtract to get — the elimination method.
Three unknowns (§613–614)
Express from all three equations, equate pairs to obtain two equations in and , then apply the two-unknown method.
Example (§613):
Expressing from each and equating: and . Then and .
Example (§614): More general 3×3 system:
Two rounds of elimination reduce to , then , .
Four unknowns — substitution chain (§621)
For cyclic systems of the form , , etc., express each subsequent variable in terms of the first unknown , propagate through all equations, and use the final equation to solve for alone. The general solution (§621) is:
with symmetric patterns for , , .
The auxiliary-variable trick (§615–616)
For problems where the sum of the unknowns appears naturally, introduce as an additional variable. This converts three coupled equations into three decoupled equations each giving one variable in terms of ; adding them recovers itself.
Q4 — Three players (§616): After three rounds of doubling-and-transfer, each player ends with 24 guineas. Setting and tracing each game:
Solving each for , , in terms of , then summing and using gives . Hence A started with 39 guineas, B with 21, C with 12.
Inverse solution (§617): The same problem can be solved backwards (without algebra) by reversing each game’s doubling step, illustrating that forward algebra and inverse reasoning give identical answers.
Military companies with sum variable (§622)
Q7 (§622): Captain with Swiss (), Swabian (), Saxon () companies; total reward 901 crowns. Each assault scenario gives one equation in one unknown and :
Multiplying to common coefficient (6) and adding gives , so . The company sizes are 265, 583, and 689.
Debt problem (§618–619)
Q5: Two debtors owing 29 pistoles jointly: and . Standard substitution gives , .
Q6: Three brothers buying a vineyard for 100 guineas, each needing a fraction of another’s money to complete the purchase — a cyclic 3×3 system solved either by the standard chain or by substituting variables sequentially (§620).
Summary of methods
| Situation | Technique |
|---|---|
| 2 unknowns | Isolate one variable from each equation, equate |
| 2 unknowns, symmetric | Add / subtract the equations directly |
| 3+ unknowns | Eliminate one variable per round until 1 remains |
| Many unknowns, sum appears | Introduce as auxiliary unknown |