Systems of Linear Equations

Summary: A system of linear equations is a collection of simultaneous first-degree equations in several unknowns; Euler develops the substitution and elimination methods, derives explicit formulas for the two-variable case, and introduces the auxiliary-sum trick to simplify complex systems.

Sources: chapter-1.4.4

Last updated: 2026-05-03

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Definition and determinacy (§605–606)

A system of $n$ linear equations in $n$ unknowns $x_1,\dots ,x_n$ has the standard form

$a_{i1}{x}_1+a_{i2}{x}_2+\cdots +a_{in}{x}_n=b_i,i=1,\dots ,n.$

The system is determinate (has a unique solution) when the equations are independent — one equation per unknown. Fewer independent equations leave the system underdetermined; contradictory equations make it inconsistent.

Two-variable case — substitution method (§607–608)

Given

$ax+by=c\text{and}fx+gy=h,$

isolate $x$ from each equation and equate the two expressions:

$\frac{c-by}{a}=\frac{h-gy}{f}.$

This yields a single equation in $y$, solved by the rules of ch1.4.2-resolution-simple-equations. Substituting back gives $x$.

General formulas (§608):

$y=\frac{ah-fc}{ag-bf},x=\frac{cg-bh}{ag-bf}.$

These are the 2×2 case of Cramer’s rule, derived here without any matrix or determinant language. (source: chapter-1.4.4, §608)

Two-variable case — elimination by addition/subtraction (§611)

When the equations have a symmetric structure, adding or subtracting them directly eliminates one variable:

$x+y=a\text{ and }x-y=b\Rightarrow 2x=a+b\Rightarrow x=\frac{a+b}{2};2y=a-b\Rightarrow y=\frac{a-b}{2}.$

This is conceptually simpler and preferred when the coefficients allow it.

Three-variable case (§613–614)

Express one variable (say $x$) from all three equations. Equating pairs of these expressions produces two new equations in just $y$ and $z$. Apply the two-variable method to those, then back-substitute.

This “round of elimination” reduces $n$ unknowns to $n-1$, and may be repeated until one equation in one unknown remains.

The auxiliary-variable (sum) trick (§615–616)

When a problem naturally involves the sum of the unknowns, let $s=x+y+z$ (or $x+y+z+\cdots$). Each equation then expresses one unknown linearly in $s$. Summing all three expressions and using $x+y+z=s$ yields a single equation for $s$, after which each unknown follows immediately.

This technique avoids the full round-of-elimination process and is especially elegant for symmetric or cyclic problems. See ch1.4.4-resolution-two-or-more-equations for the three-player game (§616) and the military companies problem (§622).

Cyclic four-variable systems (§621)

For systems of the cyclic form $u+x/a=n$, $x+y/b=n$, $y+z/c=n$, $z+u/d=n$, successive substitution expresses $x$, $y$, $z$ each in terms of $u$ alone. The last equation then gives $u$, and the pattern of the solution is:

$u=\frac{n(abcd-bcd+cd-d)}{abcd-1},$

with analogous symmetric expressions for $x$, $y$, $z$.

Connection to earlier material

The two-variable sum-difference theorem ($x=\frac{a+b}{2}$, $y=\frac{a-b}{2}$) first appears as a one-variable result in ch1.4.3-solution-of-questions (Q1–Q2) and is rederived here as a simultaneous system — illustrating how the two-variable method subsumes simpler cases.

Related pages

• ch1.4.4-resolution-two-or-more-equations
• ch1.4.2-resolution-simple-equations
• equations
• ch1.4.1-solution-of-problems-in-general
• arithmetical-progressions