Prob. 20. It is required to find a number whose square exceeds its first power by 210.

Ans. 15.

EQUATIONS OF THE SECOND DEGREE WITH MORE THAN ONE UNKNOWN QUANTITY.

(172.) An equation containing two unknown quantities, is said to be of the second degree when the greatest sum of the exponents of the unknown quantities in any term is equal to two. Thus,

and

3x2-4x+y2=25,

7xy-4x+y=40,

are equations of the second degree.

(173.) When we have given two such equations containing two unknown quantities, we may, by the methods of Art. 108-113, eliminate one of them, and obtain a new equation containing but one unknown quantity. The solution of two equations of the second degree containing two unknown quantities, generally involves the solution of an equation of the fourth degree containing one unknown quantity. Hence the principles hitherto established are not sufficient to enable us to solve all equations of this description. In some cases, however, the resulting equation is of the second degree, and may be solved by the preceding rules.

Ex. 1. Given xy=81

QUEST.-What is an equation of the second degree containing two unknown quantities? How do we solve equations of the second de gree containing two unknown quantities?

From the second equation, x=9y.

Substituting this value in the first equation, we have

Ex. 2. Given x+y:x::5:3,) to find the values of

xy=6,

}

From the first equation, we find

x and y.

Ex. 3. Given x+y=34,) to find the values of z

Adding twice the second equation to the first, we

Adding equation (a) to equation (b), we have

Whence, from equation (a), y=3.

Ex. 4. Given x+y:x::7:5,) to find the values of

xy+y=126,

From the first equation we obtain

5x+5y=7x.

x and y.

Substituting this value for x in the second equation,

Ex. 5. Given x+y:x-y:: 8:1, to find the values

xy=63, of x and y.

Ans. x=9, y=±7.

Ex. 6. Given x+y=21,) to find the values of x

and y.

Ans. x=12, y=9.

Ex. 7. Given x+y= 23,) to find the values of z

Ex. 8. Given x+y= 20, to find the values of a

Ex. 13. Given 2x2+xy-5y2=20,) to find the values 2x-3y= 1,) of x and y.

Ans. x=5, y=3.

Ex. 14. Given x2+y=281, to find the values of x

Ex. 15. Given y2+4x=2y+11, to find the values

Ans. x=2 or -46, y=3 or 15.

Ex. 16. What two numbers are those whose differ.

ence, multiplied by the greater, produces 40; and whose difference, multiplied by the less, produces 15? Let x= the greater number,

and y= the less number.

Ex. 17. What two numbers are those whose difference, multiplied by the less, produces 42; and whose difference, multiplied by their sum, produces 133? Let x the greater number,

Ex. 18. What number is that the sum of whose