ch1.4.2-resolution-simple-equations

Ch 1.4.2 — Of the Resolution of Simple Equations, or Equations of the First Degree

Summary: Euler gives systematic rules for isolating the unknown $x$ in any linear equation, handling successive complications: constant terms, fractional coefficients, $x$ on both sides, $x$ in a denominator, radical expressions, and $x$ as an exponent.

Sources: chapter-1.4.2

Last updated: 2026-05-03

---

Goal (§573)

The aim is to reduce any equation, however complicated, to the normal form $x=\text{known number}$. The chapter lays out the rules step by step, from the simplest forms upward.

Case 1 — Additive / subtractive constant (§574–576)

Equation form	Operation	Result
$x+a=b$	subtract $a$ from both sides	$x=b-a$
$x-a=b$	add $a$ to both sides	$x=b+a$
$x-a+b=c$	add $a-b$ to both sides	$x=c+a-b$

The principle: transpose any constant to the opposite side by changing its sign.

Case 2 — Multiplicative coefficient (§577)

Given $ax+b-c=d$, first clear constants ($ax=d-b+c$), then divide by $a$:

$x=\frac{d-b+c}{a}.$

Examples: $2x+5=17\Rightarrow x=6$; $3x-8=7\Rightarrow x=5$.

Case 3 — Fractional coefficient (§578–579)

Given $\frac{x}{a}=b$, multiply both sides by $a$ to get $x=ab$.

Given $\frac{ax}{b}=c$, multiply by $b$ then divide by $a$: $x=\frac{bc}{a}$.

Examples:

• $\frac{1}{2}x-3=4\Rightarrow x=14$
• $\frac{3}{4}x+\frac{1}{2}=5\Rightarrow x=6$

Case 4 — Multiple $x$-terms (§580–581)

On one side: combine into a single coefficient.

$x+\frac{1}{2}x+\frac{1}{3}x=44⟹\frac{11}{6}x=44⟹x=24.$

In general, $ax-bx+cx=d$ becomes $(a-b+c)x=d$, so $x=\frac{d}{a-b+c}$.

On both sides: move all $x$-terms to the side with more, then proceed.

$3x+2=x+10⟹2x=8⟹x=4.$

Case 5 — $x$ appears in a denominator (§582)

Multiply the entire equation by the denominator containing $x$.

Example: $\frac{100}{x}-8=12\Rightarrow \frac{100}{x}=20\Rightarrow 100=20x\Rightarrow x=5$.

Example with compound denominator: $\frac{5x+3}{x-1}=7$. Multiply by $(x-1)$:

$5x+3=7x-7\Rightarrow 2x=10\Rightarrow x=5.$

Case 6 — Radical signs (§583)

Square both sides to eliminate a square root.

Example: $\sqrt{100-x}=8\Rightarrow 100-x=64\Rightarrow x=36$.

Case 7 — $x$ as an exponent (§584)

Take logarithms of both sides.

$2^x=512\Rightarrow x\log 2=\log 512\Rightarrow x=\frac{\log 512}{\log 2}=9.$

$5\cdot 3^{2x}-100=305\Rightarrow 3^{2x}=81\Rightarrow 2x=\frac{\log 81}{\log 3}=4\Rightarrow x=2.$

This case connects linear equations back to logarithms.

Practice questions (pp. 178–179)

The chapter closes with 26 practice problems. A sample of notable ones:

No.	Problem	Answer
8	$a-\frac{b}{x}=c$	$x=\frac{b}{a-c}$
13	$\sqrt{\frac{2}{3}x+5}=7$	$x=6$
14	$x+\sqrt{a^2+x^2}=\frac{2a^2}{\sqrt{a^2+x^2}}$	$x=a\sqrt{\frac{1}{3}}$
19	$\sqrt{x}+\sqrt{a+x}=\frac{2a}{\sqrt{a+x}}$	$x=\frac{a}{3}$
20	$\sqrt{a^2+x^2}=\sqrt[4]{b^4+x^4}$	$x=\frac{\sqrt{b^4-a^4}}{2a^2}$

Related pages

• ch1.4.1-solution-of-problems-in-general
• ch1.4.3-solution-of-questions
• equations
• logarithms
• powers-and-exponents
• roots