Examples of Elimination by the method of the Greatest Common Divisor. 7. Obtain one equation with one unknown quantity from the three equations x + y + z = a, x2 + y2+x2=b, x y + x z + y z = c, by the elimination of x and y. Ans. These three equations involve an impossibility unless a2b― 2 c = 0; and in case this equation is satisfied by the given values of a, b, and c, the three given equations are equivalent to but two, one of them being superfluous, and, by the elimination of x, they give the indeterminate equation with two unknown quantities y2 + y z + z2 — ay—az+c=0. 8. Obtain one equation with one unknown quantity from the three equations Ans. 28-826 +16 24+z-10=0. 9. Obtain one equation with one unknown quantity from the four equations x+y+z+u= a, x y + x z + xu+yz+yu+zu=b, xyzu=e, by the elimination of x, y, and z. Ans. ua u3+bu2—cu+c=0. which, being substituted in the first of the given equations produces x= = 3. 11. Solve the two equations 2 x2 y1—8 y2x2+16 x2 -- 90 x y +60 (x—y2)—720 (y-1) (y2-4y+4) x 5 12 = 3 12. Solve the three equations x y + z = 5, x y z +z2 = 15, x y2+ x2 y2x+2x=8. Ans. x= 2, y = 1, z = 3. 159. Problem. To solve two equations of the first degree by Elimination by Addition and Subtraction. Solution. The given' equations may, as in art. 146, be reduced to the forms Ax+By+ M = 0, A' x + B' y + M' = 0. The process of the preceding article, being applied to these equations in order to eliminate x, will be found to be the same as to Multiply the first equation by A' the coefficient of y in the second, multiply the second by A the coefficient of x in the first, and subtract the first of these products from the second. Examples of Elimination by Addition and Subtraction. Thus, these products are AAx+A' By + A' M = 0, and the difference is (A B' — A' B) y + A M' — A' M=0; whence A' M A M y = ABA'B' In the same way y might have been eliminated by multiplying the first equation by B', and the second by B, and the difference of these products is whence (A B'A' B) x + B' M - BM' = 0; 160. Corollary. This process may be applied with the same facility to any equations of the first degree. 161. EXAMPLES. 1. Solve, by the preceding process, the two equations 13x7y34174y+431 x, 2x+y=1. Ans. x 12, y = 50. 2. Solve, by the preceding process, the two equations Examples of Elimination by Addition and Subtraction. 3. Solve, by the preceding process, the three equations 30, x + y + z = 27x+9y+3 z = 64. Ans. x = 3, y: =- 4. Solve, by the preceding process, the three equations 23x200 5 y + 360, 2y+3z548. Ans. x 360, y = 124, z=100. = 5. Solve, by the preceding process, the four equations Ans. x= 100, y = 60, z=— - 13, u—— 50. 6. Solve, by the preceding process, the four equations Ans. x= 12, y=30, z=168, u= =50. 7. Solve, by the preceding process, the six equations Examples of Elimination by Addition and Subtraction. 8. A person has two large pieces of iron whose weight is required. It is known that ths of the first piece weigh 96 lbs. less than 2ths of the other piece; and that ths of the other piece weigh exactly as much as ths of the first. How much did each of these pieces weigh? Ans. The first weighed 720, and the second 512 lbs. 9. $2652 are to be divided amongst three regiments, in such a way, that each man of that regiment which fought best, shall receive $1, and the remainder is to be divided equally among the men of the other two regiments. Were the dollar adjudged to each man in the first regiment, then each man of the two remaining regiments would receive $; if the dollar were adjudged to the second regiment, then each man of the other two regiments would receive $; finally, if the dollar were adjudged to the third regiment, each man of the other two regiments would receive $4. What is the number of men in each regiment? Ans. 780 men in the first, 1716 in the second, and 2028 in the third regiment. 10. To find three numbers such that if 6 be added to the first and second, the sums are to one another as 2:3; if 5 be added to the first and third, the sums are as 7: 11; but if 36 be subtracted from the second and third, the remainders are as 6: 7. Ans. 30, 48, 50. 10 |
