ELIMINATION BY ADDITION AND SUBTRACTION.

(79.) From what has been done, we discover that an unknown quantity may be eliminated from two equations, by the following

RULE.

Operate upon the two given equations, by multiplication or division, so that the coefficients of the quantity to be eliminated may become the same in both equations; then add or subtract the two equations, as may be necessary, to cause these two terms to disappear.

EXAMPLES.

4. Given, to find x and y, the two equations

5. Given, to find x and y, the two equations.

х

Clearing these equations of fractions, by multiplying

each by 6, they become

3x+2y=36,

2x+3y=39.

Multiplying (3) by 3, and (4) by 2, they become

9x+6y=108,

4x+6y 78.

(3)

(4)

(5)

(6)

6. Suppose we wish to find x, y, and z, from the three

We will first eliminate y: for this purpose multiply (3),

first by 4 and then by 6, and it will give

8x+4y+24=184,

12x+6y+36z=276.

(4)

(5)

Add (1) to (5); and subtract (2) from (4), and we have

We have now the two equations (6) and (7), and but

two unknown quantities x and ≈.

Multiply (7) by 17 and it will become

17x+459%=2805.

(8)

Multiplying (10) by 6, and (12) by 2, and then taking

their sum, we find

6z +2x=42.

(13)

Subtracting (13) from (3), we get

y = 4.

(80.) We will now repeat the solution of this last question, adopting a simple and easy method of indicating the successive steps in the operations.

The method which we propose to make use of, is to indicate by algebraic signs, the same operations upon the respective numbers of the different equations, as we wish to have performed upon the equations themselves.

This kind of notation will become familiar by a little

practice.

We will now resume our equations of last example.

Collecting equations (12), (16), and (10), we have

We will solve one more set of equations by this method, giving all the steps at length, the better to illustrate this

Collecting equations (33), (25), (20), (17), (29), we have