Algebra for Beginners

Substitute this value of x in the second equation, and we obtain

Then substitute this value of y in either of the given equations, and we shall obtain = 9.

Substitute this value of y in the second equation, and

we obtain

210. Third method. Express the same unknown quantity in terms of the other from each equation, and equate the expressions thus obtained.

Thus, taking again the same example, from the first

100-7y

equation x=

and from the second equation

2 8

Clear of fractions, by multiplying by 24; thus

From this equation we shall obtain x=9; and then, as before, we can deduce y=4.

211. Solve

19x-21y=100, 21x-19y=140.

These equations may be solved by the methods already explained; we shall use them however to shew that these methods may be sometimes abbreviated.

Again, from the original equations, by subtraction, we

obtain

that is, therefore

21x-19y-19x+21y=140-100;

2x+2y=40;

x+y=20.

Then since x-y=6 and x+y=20, we obtain by addition 2x=26, and by subtraction 2y = 14;

212. The student will find as he proceeds that in all parts of Algebra, particular examples may be treated by methods which are shorter than the general rules; but such abbreviations can only be suggested by experience and practice, and the beginner should not waste his time in seeking for them.

If we cleared these equations of fractions they would involve the product xy of the unknown quantities; and thus strictly they do not belong to the present chapter. But they may be solved by the methods already given, as we shall now shew. For multiply the first equation by 3 and the second by 2, and add; thus