x 4. Given 3—4 +6, to find the value of x. 2 Here (10), multiplying the equation by 2, the denominator of the fractional part, the equation becomes 6x-8=x+12; .. by transposition, 6x-x=12+8; then, Case II. Addition, 5x=20; .'. (by Axiom 4,) x= 3x 20 =4. 5. Given 4x+=+16, to find the value of x. 5 3 Here (10), multiplying the equation by 15, the least common multiple of 5 and 3, 60x+3x=15x +240; .. (6.) by transposition, 60x+3x-15x=240; then, by Addition, 48x=240; 6. Given 4ax+6ab=2cx+7a, to find the value of x Here (6), by transposition, 4ax-2cx-7a-6ab; ... (Ax. 4.) x= 7. Given 7a-6ab 4a-2c x 4 8 + + = +4, to find the value of x. a b C d Here (10), multiplying the equation by abcd, the product of the denominators, it becomes, bcdx+acdx+4abd=8abc+4abcd; by transposition, bcdx+acdx=8abc+4abcd-4abd; Here (10), multiplying the equation by 15, the least common multiple of 3 and 5, 60x+9x-12=105+5x+75; by transposition, 60x+9x-5x=105+12+75, or 64x=192; .'. (Ax. 4,) x= 3. 12x-12 3x+1 10x+6 192 64 9. Given 42+ + +15, to find 2 the value of x. Here, multiplying the equation by 16, the least common multiple of 16, 8, and 2, the equation becomes, 672+12x-12=6x+2+80x+48 +240; by transposition, 672-12-2-48-24080x+6x-12x, or 370=74x; x-2 10. Given 3x- =16, to find the value of x. Multiplying the equation by 2, 6x−x+2*=32; by transposition, 6x-x=32-2; or 5x=30; 30 NOTE. It may not be improper to remark why the 2, with the asterisk over it, becomes +, when cleared of fractions. The reason is obvious, for in clearing the equation of the fraction, the x, which is +, becomes minus, therefore, the 2 must be made plus in order to obtain the difference between the x and the 2, as required in the given equation. It may also be further remarked, that if any fraction, hav ing a compound quantity for its numerator, be a minus quantity, all the signs in the numerator must be changed when the equation is cleared of fractions, as in the eleventh and twelfth examples. Here (10), multiplying the equation by 6, the least common multiple of 3, 2, and 2, 24x-2x-8+2=93-9x-6; by transposition, 24x-2x+9x=93+8—2—6, Here, multiplying the equation by r, 3rx-ax-b+c-d=nr; by transposition, 3rx—ax=nr+b—c+d; Multiplying this equation by 2y—8, 63y-117-90y-360; by transposition, 360-117-90y-63y, Multiplying this equation by 5y-12, by transposition, 144y+25y=414+60, or 10y+4=36-2y; by transposition, 10y+2y=36—4, Here, by Theo. 2, (3+4+4)2=("+7—2).12, .. multiplying by 5, 6y+8=20y-20; by transposition, 8+20=20y-6y; or 28-14y; 17. Given √9y+13=7, to find the value of y. Here (13), squaring each side of the equation, 18. Given √4y+36—2—2, to find the value of y. Here (Cor. to 13), √4y+36=2+2=4; y+988+2=6, to find the value of 5 Here (Cor. to 13), √y+988—6—2=4; y. ... (13) by raising each side to the 5th power, y+988=1024; by transposition, y=1024-988-36. 20. Given √/4y—11=2/7—1, to find the value of y. Here, squaring each side of the equation, 4y-11-4y-4√√y+1 then (by 7, and transposition,) 4y=11+1=12; |