Elements of Algebra

CHAP. III. OF ALGEBRA. 317 we obtain $y = 124 - \frac{14z}{5}$: so that $z$ must be divisible by 5. 5. If therefore we make $z=5u$, we shall have $y = 124 - 14u$; which values of $y$ and $z$ being substituted in the first equation, we have $3x - 35u + 620 = 560$; or $3x = 35u - 60$, and $x = \frac{35u}{3} - 20$; therefore we shall make $u = 3t$, from which we obtain the following answer, $x = 35t - 20$, $y = 124 - 42t$, and $z = 15t$, in which we must substitute for $t$ an integer number greater than 0 and less than 3: so that we are limited to the two following answers: If $t=1$, we have $x=15, y=82, z=15$. If $t=2$, we have $x=50, y=40, z=30$. --- CHAP. III. Of Compound Indeterminate Equations, in which one of the Unknown Quantities does not exceed the First Degree. 31. We shall now proceed to indeterminate equations, in which it is required to find two unknown quantities, one of them being multiplied by the other, or raised to a power higher than the first, whilst the other is found only in the first degree. It is evident that equations of this kind may be represented by the following general expression: $a+bx+cy+dx^2+exy+fx^3+gx^2y+hx^4+kx^3y+\ldots = 0$. As in this equation $y$ does not exceed the first degree, that letter is easily determined; but here, as before, the values both of $x$ and of $y$ must be assigned in integer numbers. We shall consider some of those cases, beginning with the easiest. 32. Question 1. To find two such numbers, that their product added to their sum may be 79. Call the numbers sought $x$ and $y$: then we must have $xy+x+y=79$; so that $xy+y=79-x$, and $y = \frac{79-x}{x+1} = -1 + \frac{80}{x+1}$, by actual division, from which we see that $x+1$ must be a divisor of 80. Now, 80 having