clusions, unless some of them be dependent. The number of equations deducible from others, that is, dependent upon them, must just be equal to the excess of the number of the equations above that of the unknown quantities, in order that the latter may have determinate values. Thus, if the number of unknown quantities be two, and of equations five, then three of the latter must be dependent on some of the other equations, and these three being unnecessary, and therefore rejected, the remaining two will determine the values of the two unknown quantities.

(266.) When the number of given equations exceeds that of the unknown quantities, if the latter be eliminated, the result will be equations containing only given quantities. If these equations be consistent, it is a proof that some of the given equations are dependent, and therefore unnecessary; if they be inconsistent, this would imply an impossibility of fulfilling the given equations by any values whatever of the unknown quantities; but if the given quantities are disposable, that is, if it be allowed to change their values arbitrarily, then these resulting equations may be all fulfilled by assigning proper values to these quantities. These values being properly adjusted, the said equations will be the conditions that must be fulfilled, in order that all the given equations be consistent, and the values of the unknown quantities determinate. Hence these equations among the given quantities are called equations of condition.

To illustrate the observations in the last paragraph, let there be given two equations containing only one unknown quantity, thus,

[3]

and a being thus eliminated, the last equation among the given quantities is the equation of condition that they must

fulfil, in order that a value of a may be capable of satisfying the given equations. That condition ad be may also be expressed thus (357),

a:b-c:d,

or these four quantities must be proportional. If they are not, the equation [3] cannot exist, and therefore it would imply an absurdity.

If there are given three equations containing only two unknown quantities, the latter being eliminated, the resulting equation among the known quantities will be the equation of condition. A value of the two unknown quantities being found from two of the given equations, these quantities may be eliminated from the other, by substituting Thus, let the three equations ax + by=p

these values for them.

cx + dy = q

mx + ny =r

be given, to find the equation of condition.

The values of x and y may be found from the first two, as in (263), which are

and these values being substituted in the third, give

or

m (dp — bg) n (aq―cp)

+

ad bc

ad bc

=r

m (dpbq)+n (aq―cp)+r (bc―ad)=0 and unless the values of the given quantities be such as to verify this last equation, or to make its first member = 0, no values of the unknown quantities could be found to satisfy the three equations. If, however, this equation be fulfilled, the values of x and y will then satisfy the three equations; but these values being found from the first two, the third is unnecessary.

1. The sum of two numbers is 60, and their difference is 12: what are these numbers?

These numbers verify the conditions of the question, for x+y=36+ 24 = 60, the sum xy=36—24=12 difference.

...

2. The sum of two numbers is 44, and their ratio is that of 5 to 6: required the numbers.

3. The sum of two numbers is 16, and the sum of their reciprocals is double of their difference: what are the num

multiply [2] by xy, y+x=2y-2x

4. The sum of two numbers is 10, and twice the less is to three times the greater, as the square of the less to that of the greater: required the numbers.

and is the same as the difference between the x and y considered as positive; and this results from the principle, that to subtract a quantity is just adding it with an opposite sign.

8. The values of the properties belonging to two persons are as m to n, and the difference of their values is a: required the values of their respective properties.

Let the value of greater property,

Then as a = the difference of their properties, the solution will lead to exactly the same result as in the last question; therefore

If while a is the greater, m should ben, then ≈ and y are negative, and the negative value of x is less than that of y (55). Instead of x and y then representing absolute property, they represent only negative property, or debt. If an amount of property a be added to the negative property y, the sum will be equal to the negative property x, for

where x and y are negative, because (m—n) is so, as m▲n

9. A's age is double of B's, but 15 years ago A's was 5 times B's age: required their ages.

10. A bill of £210 was paid in sovereigns and crowns, and the number of the latter used was three times that of the former: how many pieces of each were used?