Ex. 3. Let x + N,/ (5 x + 10) = 8; to find x.` By transposition, .„/ (5 x + 10) = 8 —a. squar. both sides 5 x + 10 = 64 — 16.r + x2 — 21 x = 10 — 64 —54 complete 441. 441 the sq., X2 — 21 x + —4— — 54 4 441 —216 441 225 , or x2-21 x + = 4 4 4 21 15 extracting the sq. root, x — * x 21 ± 15 — 3 or 18. 2 By this, process two values of x are found, but ontrial it appears, that 18 does not answer the condition of the equation, if we suppose that ‘,/ (5 x + 10) repre sents the positive square root of 5 x + 10. The reason is, that 5 x + 10 is the square of — (5 ± 10) as well as of + .„,/ (5 x + 10); thus by squaring both sides of the equation (5 x + 10) 8 — x, a new condition is introduced, and a new value of the unknown quanti ty corresponding to it, which had no place before. Here, 18 is the value which corresponds to the supposition that x — A/ (5 x + 10) 8.
Every ectuation, where the unknown quantity is found in two terms, and its in dex in one is twice as great as in the other, may be resolved in the same manner.
Ex. 4. Let ;1-4 il=21 z. +4 z1-1-4=21 --I- 4=25 z +2=-±5 z i=±5 —2 = 3, or — 7 therefore z=9, or 49.
Ex. 5. Let y4 — 6y1— 27=0.
y4 — 6e=27 y4 — 6y,+9---27+ 9 = 36 y' — 3y= ±6 y' — 3 *6=9, or — 3 ±3, or ± ‘,/ — 3.
• Ex. 6 Let. y6-Fr y3 + 75'73 0.
q3 e+, y3 — 74 r' V5:116+1' !"13 71,^ -97 ..3, r' ce3 5 7-2 — hot 4 F7 ) • y3= - I 1"__ 93 \ 2 4 27) yi Y= 7[--;±3/4/( 4 27 ) When there are more equations and unknown quantities than one, a single equation, involving only one of the un known quantities, may sometimes be ob tained by the rtdes laid down for the so lution of simple equations ; and one of the unknown qitantities bei ng discovered, the others maybe obtained by- substituting its value in the preceding equations.
Ex.7. Let / x9-Y2 =65 ?• To find x andy. x y=2,8 From the second equation, 2 x y=56 &adding this to the 1st, x'+2xy+e=121 sub. it from the same, x2 — 2.ry+ys=_-9 by extracting the sq. roots, x+y= ± 11 and x — y= ± 3 t_herefore, 2 x= ± 14 x=7, or —7 and y=4, or —4