Cor. Hence, any quantity may be multiplied and divided by any other quantity without altering its value. 5. Things which are equal to the same things, are equal to each other. 5. In the solution of equations all the unknown quantities must be transposed to one side of the equation, and all the known quantities to the other; then proceed as in Addition. Thus, in the equation 3x+4=a+x, the 3x and x are considered unknown quantities, and the 4 and the a are known quantities. (See the transposition in the next definition.) 6. When any quantity is transposed from one side of the equation to the other, its sign must be changed, Thus, if 3x+7=28; by transposing the known quantity to the other side of this equation it becomes 3x=28-7, then 3x=21; therefore (axiom 4) x= 7. Hence, transposing the 7, in this case, is only subtracting it from each side of the equation. Thus, if 7 be subtracted from 3x+7, 3x will remain, and if 7 be subtracted from 28, 21 will remain; then (axiom 2), 3x=21; and (axiom 4), x=7. Also, if 3x+4=a+x; then by transposition 3xa+x x=a—4, or 2x=a-4; ... (axiom 4) x=~ 2 Also, if 3x-4=2x+3, then by transposition, 3x-2x=3+4, or x=7. In this case, transposing the 4 is the same as adding 4 to each side of the equation, and transposing the 2x is subtracting it from each side of the equation. Cor. Hence, if all the signs in an equation be changed, the two sides will remain equal, because in this case every term is transposed. 7. If the same or equal quantities be found on each side of the equation, with like signs, they may be omitted in the transposition. Thus, if 5x+3=4a+3, here 3 is found with the same sign on each side of the equation, then will 5x-4a, (3 being omitted in the transposition) and (bý axiom 4), x== 4a 5 8. If every term in an equation be multiplied by any number, the equation will retain its equality, (axiom 3). Thus, if 3x+4=16, multiply the whole by x, and it becomes 3x2+4x=16x. 9. If every term in any equation be divided by the same number, the equation will retain its equality (axiom 4). Thus, if 3x2+4x=16x; here, a being common to each term, the whole may be divided by x, and the equation becomes 3x+4=16. 10. If any of the terms in an equation be a fraction, the equation may be cleared of the fractional part by multiplying the whole equation by the denominator of the fractional part; or if there be two or more fractional parts, the whole equation must be multiplied by the product of the denominators, or their least common multiple. Thus, if 3x+13, multiplying the whole of this equation by 4, and it becomes 12x+x=52, or 13x=52;... (axiom 4)x=4. 2x x 3 4 6 4 Also, if+ -+ 13; multiply the whole of this equation by 72, the product of all the denominators, and it becomes 48x+18x+12x=936, or 78x=936; ... x=12. But if the whole equation be multiplied by 12, the least common multiple of 3, 4, and 6, it becomes 8x+3x+2x=156, or 13x=156, .·.x=12. Also, if + = ; multiplying every term of this equation by a b c, the product of the denominators, it becomes bcx+a2c2=a by. 20 ac y a b' C 11. If each side of an equation be squared, cubed, or raised to any other power, the results will be equal. Thus, if x 4, then by squaring each side, a2=16, or by cubing each side, a=64, and so on for any other power. D Also, if x+y=4, then by squaring each side, x2+2xy+y=16; or, by cubing each side, 2+3x2y +3.xy2+y3=64, &c. 12. If equal roots be extracted on each side of any equation, the roots will be equal. Thus, if x2=16, extracting the square root of each side, x=4. If x64, extracting the cube root, x=4. If x2+2xy + y2 =16, extracting the square root of each side, x+y=4, and so on for any other power, being only the reverse of Def. 11. ་ ན་ 13. If one side of an equation be an irrational or surd quantity, it may be taken away by raising each side to the power denoted by the radical sign. Thus, if x+4=7; this signifies that the square root of x+4 is equal to 7; squaring each side, x+4=49, or which is the same thing, √x+4x√√x+4=72 x+42×x+4*2=x+4x+4; therefore x+4=49, (by Note 2. Case I. Multiplication). or Cor. But if a part only of one side of an equation be a surd quantity, it may be taken away by transposing the rest of the quantities to the other side of the equation, and then raising each side to the power denoted by the radical sign. 3. Thus, √x+4+2=4; this expresses, that the cube root of x+4 added to 2, is equal to 4. transposition +4=4-2=2; cubing each side of this equation (it being the power denoted by the sign), x+4=8; that is, By 3 3 13 √x+4× √x+4x√x+4=8. In order to make this a little clearer, x+4 being the same as x+43, then by Note 2. Case I. Multiplication, x+438, the cube or third power, being = ; as above. 5 x+43=x+4; whence x+4=8, 5 Also, if √x+2=3;..√x+2 being =x+2| x+25=3, x+25=3, x+29, the square, x+2|3=9, x+25=81, the fourth power, x+25=3, x+25=243 the fifth power, as above, x+25=x+2; whence x+2=243. Irrational or surd quantities are such as have no exact root, the root being extracted by placing over the quantity a radical sign or fractional index. Thus, the radical sign✔ for the square root,✔ for the cube root, &c. or for the square root, for the cube root, &c. therefore 3 or 3 is called an irrational number or surd quantity, because no number, either whole or fractional, can be found, which when multiplied into itself, will produce 3. This being the case with x+4, the surd part of the equation mentioned in the Definition, . 14. In some cases irrational or surd quantities require to be reduced into different forms before they can be used. This may be done by resolving or decomposing the surd into two factors, and extracting the root of that which is rational. Thus, √639 x√7 (by Note to Case III. in Multiplication), extracting the square root of 9, which is rational, the expression becomes 3 x 7, or 3/7; also, /24a3x3 will resolve into 4a2x2× √√6ax; extract the square root of 4a2x2, the expression becomes 2ax × √6ax, or 2ax/6ax. 3 3 Also, 54a3rty will resolve into √27a3x3 × √2xy, extract the cube root of 27a33, the expression becomes 3ax √2xy; also,ab will resolve into √2 × b; extract the square root of a2, the expression 4,16α x6+16a6x7 Also, ✩ a x2 + a 2 x 3 bc2+242x 81bc2+x16266x will resolve into 16q+x+ ; extracting the fourth root of 8164 X (as Cor. It is evident from the above, that before any quantity can be multiplied into a surd quantity, and the product placed under the radical sign, the quan |