Lesson 5 · from §72 of the 1910 text
Literal Equations
Literal equations
A literal equation is one in which some or all of the known numbers are letters. The rule is the same; the only new step is that, after collecting, you may need to factor out x and divide by the factor.
Example 1. Solve ax + b^2 = a^2 + bx for x.
Transpose: ax - bx = a^2 - b^2. Factor: x(a - b) = (a - b)(a + b). Divide by a - b: x = a + b. Check: a(a + b) + b^2 = a^2 + ab + b^2, and a^2 + b(a + b) = a^2 + ab + b^2. ✓
Example 2. Solve \dfrac{x}{a} - 1 = \dfrac{x}{b} for x.
Multiply by ab: bx - ab = ax. Transpose: bx - ax = ab, so x(b - a) = ab and x = \frac{ab}{b - a}.
Type the answer using the other letters: a + b, ab/(b - a). Any equivalent form is accepted.
Try one: solve \dfrac{x}{a} - 1 = \dfrac{x}{b} yourself.
Quick check
\displaystyle \frac{x}{a}-1=\frac{x}{b}
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Solve for x. Type the root in terms of the other letters, like x = a + 2b or 3c/4.
\displaystyle 5cx-8c^2=4c^2-cx
\displaystyle 2(x+1)-4k=2
\displaystyle 3(2x-a)=2(x-2a)
\displaystyle ax+bx=a^2+ab
\displaystyle cx+b^2=bx+bc
\displaystyle mx+n^2=m^2-nx
\displaystyle 6ac+cx+4a^2=2ax+3c^2+2ca
\displaystyle 5ax-5a^2+6b^2=7ab+3bx
\displaystyle \frac{x}{2a}=b
\displaystyle \frac{3ab}{x}=a
\displaystyle \frac{a}{x}+\frac{3a}{2x}=\frac{5}{4}
\displaystyle \frac{4a}{3x}+\frac{4a}{x}=\frac{3}{2}+\frac{5a}{6x}
\displaystyle \frac{x}{a}+\frac{x}{b}=a+b
\displaystyle \frac{c^2}{x}-c=\frac{d^2}{x}+d
\displaystyle \frac{ax}{2b}-4b^2=\frac{2bx}{a}-a^2
\displaystyle \frac{x}{c}+\frac{c-2x}{3}-3c=-4
\displaystyle \frac{x}{a}+\frac{x}{c}+ac=bc+ab+\frac{x}{b}
\displaystyle \frac{3b+4x}{5b}+\frac{3b+2x}{4b}=\frac{1}{20}
\displaystyle \frac{ax}{2}-\frac{3b}{5}\left(x-\frac{2ab}{3}\right)=ab\left(\frac{a}{2}-\frac{b}{5}\right)
\displaystyle \frac{2x-3b}{a}+\frac{2}{b}\left(\frac{3x}{2}-a\right)+5+\frac{9b}{a}=\frac{4a}{b}
\displaystyle \frac{x-m^2}{x-n^2}=\frac{n}{m}
\displaystyle \frac{x}{c}-\frac{d}{c}=\frac{x+2c}{d}-3
\displaystyle \frac{b}{b(b-x)}+\frac{3}{a(b-x)}+\frac{3a+9}{2ab}=0
\displaystyle \frac{c}{a(x+c)}+\frac{a}{c(x-a)}=\frac{c^2-ac+2a^2}{2ac(x-a)}
\displaystyle \frac{1}{ab}+1-\frac{ab}{x}+\frac{1}{abx}=0
\displaystyle \frac{a^2}{bx}+\frac{b^2}{ax}=\frac{1}{a}+\frac{1}{b}
\displaystyle \frac{c^2}{dx}-\frac{d^2}{cx}-\frac{3c-3d}{x}=\frac{c-d}{cd}
\displaystyle \frac{a+x}{b+x}-\frac{a-x}{b-x}=\frac{2}{x^2-b^2}
\displaystyle \frac{\frac{x}{a-b}-a}{\frac{x}{a-b}+a}+1=\frac{2}{a^2+ab+1}
\displaystyle \frac{\frac{a}{2}+\frac{x}{3}}{\frac{a}{3}-\frac{x}{2}}+\frac{7a}{4a-6x}=-7
\displaystyle \frac{a^2+ac}{x+3c}+\frac{2cx(a+c)}{x^2+5cx+6c^2}=\frac{a^2+2ac+c^2}{x+2c}