Page:Elementary algebra (1896).djvu/175

157 Rh LITERAL SIMULTANEOUS EQUATIONS.

177. Ex. 1. Solve ax+by = c (1), a'x+b'y = c' (2).

The notation here first used is one that the student will frequently meet with in the course of his reading. In the first equation we choose certain letters as the coefficients of x and y. and we choose corresponding letters with accents to denote corresponding quantities in the second equation. There is no necessary connection between the values of a and a', read " a and a prime,'" and they are as different as a and b ; but it is often convenient to use the same letter this slightly varied to mark some common meaning of such letters, and thereby assist the memory. Thus a and a' have a common property as being coefficients of x: b, b' as being coefficients of y.

Sometimes instead of accents letters are used with a suffix such as a_1, a_2, a_3, b_1, b_2, b_3 etc., read " a sub one, a sub two." etc.

To return to the equation ax+by = c (1), a'x+b'y = c' (2).

Multiply (1) by b' and (2) by b. Thus ab'x +bb'y = b'c a'bx + bb'y = bc' ; by subtraction, (ab' - a'b)x = b'c - bc' ; x = {b'c - bc'} {ab' - a'b} (3).

As previously explained in Art. 171, we might obtain y by substituting this value of x in either of the equations (1) or (2) ; but y is more conveniently found by eliminating x, as follows :

Multiplying (1) by a' and (2) by a, we have aa'x + a'by = a'c, aa'x + ab'y = ac'; by subtraction, (a'b - ab') y = a'c - ac' ; y = {a'c - ac'}{ a'b-ab' }