Lesson 1 · from §40 of the 1910 text
Equations with Letter Coefficients
The most general simple equation in one unknown has letters as the coefficients of the unknown. Solving one often means dividing polynomials.
Example. Solve ax + 4a = a^2 + 2x + 4 for x. \begin{array}{rll} ax - 2x &= a^2 - 4a + 4 & \text{transpose} \\ (a - 2)x &= a^2 - 4a + 4 & \text{write the coefficient of } x \text{ as a binomial} \\ x &= \dfrac{a^2 - 4a + 4}{a - 2} = a - 2 & \text{divide by the coefficient} \end{array} Check: put a - 2 for x. The left side is a(a-2) + 4a = a^2 + 2a, and the right side is a^2 + 2(a-2) + 4 = a^2 + 2a. ✓
Quick check
Solve for x: bx - 3b = 2b
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 and check.
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