NOTE. The letters a', b' are read a prime, b prime. In like manner, a′′, a ́ ́ are read a second, a third, and a1, a2, as are read a sub one, a sub two, a sub three. It is sometimes convenient to represent different numbers that have a common property by the same letter marked by accents or suffixes. Here a and a' have a common property as coefficients of x.

ax + by = c.

a'x + b'y = c.

To find the value of y, multiply (1) by a' and (2) by a.

To find the value of x, multiply (1) by b′, and (2) by b.

(1)

(2)

195. Fractional simultaneous equations, of which the denominators are simple expressions and contain the unknown numbers, may be solved as follows:

Multiply (1) by 15, the L. C. M. of 3 and 5; and (2) by 30, the

Substitute the value of x in (1), and we obtain

y

(3)

(4)

196. If three simultaneous equations are given, involving three unknown numbers, one of the unknowns must be eliminated between two pairs of the equations; then a second unknown between the two resulting equations.

Likewise, if four or more equations are given, involving four or more unknown numbers, one of the unknowns must be eliminated between three or more pairs of the equations; then a second between the pairs that can be formed of the resulting equations; and so on.

NOTE. The pairs chosen to eliminate from must be independent pairs, so that each of the given equations shall be used in the process of the eliminations.

Eliminate z between the equations (1) and (3).

Eliminate z between the equations (1) and (2).

We now have two equations (5) and (6) involving two un