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).

Multiply (1) by 7,

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

14x-21 y + 28 z = 28

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

CHAPTER XII.

PROBLEMS INVOLVING TWO OR MORE UNKNOWN NUMBERS..

197. It is often necessary in the solution of problems to employ two or more letters to represent the numbers to be found. In all cases the conditions must be sufficient to give just as many equations as there are unknown numbers employed. If there are more equations than unknown numbers, some of them are superfluous or inconsistent; if there are fewer equations than unknown numbers, the problem is indeterminate.

EXERCISE 77.

1. If A gave B $10, B would have three times as much money as A. If B gave A $10, A would have twice as much money as B. How much has each?

x = the number of dollars A has,
y = the number of dollars B has.

Let

and

Then, if A gave B $10,

X

10
the number of dollars A would have,
y + 10 = the number of dollars B would have.
.. y + 10 = 3 (x - 10).

(1)

If B gave A $10,

x + 10 = the number of dollars A would have, y-10 = the number of dollars B would have. .. x + 10 = 2 (у — 10).

(2)

From the solution of equations (1) and (2), x = 22, and y = 26. Therefore, A has $22, and B has $26.

2. If the smaller of two numbers is divided by the greater, the quotient is 0.21, and the remainder 0.0057; but if the greater is divided by the smaller, the quotient is 4 and the remainder 0.742. Find the numbers.

Therefore, the numbers are 4.78 and 1.0095.

3. If A gave B $100, A would then have half as much money as B; but if B gave A $100, B would have one third as much as A. How much has each?

4. If the greater of two numbers is divided by the smaller, the quotient is 7 and the remainder 4; but if three times the greater number is divided by twice the smaller, the quotient is 11 and the remainder 4. Find the numbers.

5. If the greater of two numbers is divided by the smaller, the quotient is 4 and the remainder 0.37; but if the smaller is divided by the greater, the quotient is 0.23 and the remainder 0.0149. Find the numbers.

6. If A gave B $5, he would have $6 less than B; but if he received $5 from B, three times his money would be $20 more than four times B's. How much has each?