Elements of Algebra

206 ELEMENTS SECT. IV. feet broader and 2 feet longer, it would then have been 68 square feet larger: required the length and breadth of the floor. Ans. Length 14 feet, and breadth 10 feet. 25. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3; but two of the greyhound's leaps are as much as three of the hare's: how many leaps must the greyhound take to catch the hare? Ans. 300. CHAP. IV. Of the Resolution of two or more Equations of the First Degree. 605. It frequently happens that we are obliged to introduce into algebraic calculations two or more unknown quantities, represented by the letters $x$, $y$, $z$: and if the question is determinate, we are brought to the same number of equations as there are unknown quantities; from which it is then required to deduce those quantities. As we consider, at present, those equations only, which contain no powers of an unknown quantity higher than the first, and no products of two or more unknown quantities, it is evident that all those equations have the form $az + by + cx = d$. 606. Beginning therefore with two equations, we shall endeavour to find from them the value of $x$ and $y$: and, in order that we may consider this case in a general manner, let the two equations be, $ax + by = c$; and $fx + gy = h$; in which, $a$, $b$, $c$, and $f$, $g$, $h$, are known numbers. It is required, therefore, to obtain, from these two equations, the two unknown quantities $x$ and $y$. 607. The most natural method of proceeding will readily present itself to the mind; which is, to determine, from both equations, the value of one of the unknown quantities, as for example $x$, and to consider the equality of those two values; for then we shall have an equation, in which the unknown quantity $y$ will be found by itself, and may be determined by the rules already given. Then, knowing $y$, we shall have