Lesson 3 · from §107 of the 1910 text
Dividing One Equation by the Other
Dividing one equation by the other
Sometimes an equation simpler than either of those given can be derived from a system by dividing one equation by the other, member by member.
Example. Solve \begin{cases} x^2 - y^2 = 9 & (1) \\ x - y = 1 & (2) \end{cases}
Dividing (1) by (2), since x^2 - y^2 is the product (x - y)(x + y),
x + y = 9 \qquad (3)
The system (2) and (3) is equivalent to the system (1) and (2), and is more easily solved than the latter: adding, 2x = 10, so (x, y) = (5, 4).
Division works when one member of an equation is a factor of the corresponding member of the other, as x - y is a factor of x^2 - y^2, or x + y of x^3 + y^3. (Dividing by an expression is allowed only when it is not zero; here x - y is 1.)
Some exercises below use other letters. Give the pair in the order named in the exercise, for example (R, h).
Typing your answers. Give every set of roots as a pair in parentheses, in the order the exercise names, for example (3, 2), (-2, -3). The pairs may come in any order. You may also write x = 3, y = 2; x = -2, y = -3. Radicals are typed sqrt(3), so 2\sqrt3 is 2sqrt(3).
If a system turns out to have infinitely many sets of roots, type infinitely many.
Quick check
Solve by division: x^2 - y^2 = 24, x + y = 6.
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 (using division where possible), pair results, and check.
\begin{cases} x^{2} - y^{2} = 16 \\ x + y = 8 \end{cases}
\begin{cases} \dfrac{1}{x^2} - \dfrac{1}{y^2} = 5 \\[4pt] \dfrac1x - \dfrac1y = 1 \end{cases}
\begin{cases} R^2h - 50 = 0 \\ Rh = 10 \end{cases} Give each set as (R, h).
\begin{cases} h^2k - 100 = 0 \\ k^2h + 80 = 0 \end{cases} Give each set as (h, k).
\begin{cases} P(1 + r)^2 = 112.36 \\ P + Pr = 106 \end{cases} Give each set as (P, r).
\begin{cases} \dfrac{gt^2}{2} = .0256 \\[4pt] gt = 1.28 \end{cases} Give each set as (g, t).
\begin{cases} x^2y^2 + 6 = 5xy \\ xy + 4 = 6 \end{cases}
\begin{cases} x^{4} = y^{4} + 175 \\ x^{2} = y^{2} + 7 \end{cases}
\begin{cases} x^{2} - x y - 6 y^{2} = 16 \\ x - 3 y = 1 \end{cases}
\begin{cases} x^{2} + 7 = 4 x y + 5 y^{2} \\ x^{2} = 1 + y^{2} \end{cases}
\begin{cases} x^{3} + y^{3} = 12 \\ x + y = 2 \end{cases}
\begin{cases} x^{2} - x y + y^{2} = 6 \\ x^{3} + y^{3} = 12 \end{cases}
\begin{cases} x + y = 2 \\ x^{2} - x y + y^{2} = 6 \end{cases}