4>C 



MATHKMATIOS. ALGBBRA. 



[SIMPLE EQUATIONS. 



tioiu, except to remark, that the two instance* given 

 before, to *how the meaning of tlie bnr-niKir'nm, suggest 

 two arithmetical theorems, hi<-!>, although very iui pie, 

 .ill ilo well to remember, as they are useful. The 

 expressions referred to are 



+ 6 + (a b) -2a, ando + 6 (a 6) - 26. 

 Now, as a and b may stand for any two numbers 



whatever, we learn from these results that 



1 Tlif turn (a-)- 6) of any two numbers, increased by 

 their different* (a 6), is twice the greater number, that 

 . 



TKt rim of any two numbers, diminished by their 

 d(tfcmee, it twice the lest ; viz., 26. 



Take, for instance, tlio tiro numbers 7 and 3; their 



sum 10, increased by thoir difference 4, is 14 double of 



the : And the same sum, 10, diminished l.y 



difference, 4, is 6 double of the less; aiid the 



saiuu of any pair of numbers whatever. 



SIMPLE EQUATION'S. 



We sluill now show a few applications of the princi- 

 ples laid down in the foregoing articles to the solution of 

 simple equations. \Yhat we here propose to give, must 

 be regarded, however, as only an introduction to the 

 complete consideration of the subject, which will be 

 resumed hereafter. 



An equation is merely a statement, in the characters 

 of algebra or arithmetic, that two quantities are equal: 

 thus, that two and five are equal to seven, is a statement 

 of equality, which, when expressed in figures and signs. 

 thus, 2 + 5=7, is an equation. In like manner, all these 

 are equations namely : C + 2 3 = 5, 7 4 = 6 3, 

 2 + 8 1 = 9, Sx 2 = 10, ic. , *c. All but the last arc- 

 purely arithmetical equations; the last Ls an altjebi 



ion, as it contains the algebraical character or 

 symbol x. You see that an equation consists of two 

 members or tiiltt : one on the left of the sign of equality, 

 and the other on the right. These members or sides, if 

 equally increased, or equally diminished, equally multi 

 plied, or equally divided, give results that must evidently 

 be new equations, because equal quantities thus operated 

 upon must give equal quantities for the results of th 

 operations. This is a general axiom which you must 

 pay attention to, as your success in solving the equations 

 about to bo given will depend chiefly upon your skill in 

 applying it. But we must tell you what is meant by 

 s-Ariiig an equation. 



You are said to have solved a question in arithmetic 

 when you have worked it out and arrived at the answer; 

 till you have completed the calculation, the answer re- 

 mains unknown. It is the same in algebra ; the solution 

 of an equation is the finding the value or interpretation 

 of some letter or letters, which at the outset is unknown. 

 Unknowti values are usually represented by letters 

 towards the end of the alphabet, as z, y, x, itc. ; while 

 letters whoso numerical value are already known, are 

 chosen from the beginning, as a, b, e, <tc. In the equa- 

 tion given above, namely, 3z 2 = 10, x represents a 

 number, at present unknown, such that three times that 

 number, diminished by 2, is equal to 10; the operation 

 by which the number, or value of x, is discovered, is the 

 solution of the < This is ctfected as follows : 



Add 2 to each side ; the result is the new equation 

 8z 12 ; that in, 3 times is equal to 12. 



ide each side of this by 3: the result is x 4: so 

 that x, at first an unknown value, turns out to be 

 to 4 ; and you see that three times 4 diminished by 2 is 

 equal to IV 



As here, so in the examples which follow, x will 

 always bo used to stand for the value at the outset un- 

 known ; so that the solution of the equation will be the 

 finding the value of x ; the operations for this purpose 

 will be very easy: they are called lYonifKMmMI ami 

 Cleai 'it*. 



Tra>:*porition. The operation called transposition con- 

 sists simply in taking quantities from one side of an 

 in, .in.) (.iiiting them on the other side, still, how- 

 ever, taking care to preserve the equality of the two 



sides ; for whatever operations we perform, we most 

 never disturb the equality of the two sides : the result 

 of each operation must still be an equation. Now, if 

 you remove a quantity from one side of an equation, 

 and place it on the other side ; that is, if you rub it out, 

 sign and all, from one side, and then write it down on 

 the other, the change you thus make will not disturb the 

 equality of the two sides, provided only the quantity 

 rubbed out be written on the other side with a changed 

 sign. You will bo convinced of this by an example or 

 two. Let there bo the equation 4x 5 3x 2 : then 

 if we wish the 5 on the left to bo removed to the right, 

 all wo have to do is to add 6 to buth sides of the equa- 

 tion; the result is the equation 4.c = 3x 2 + 5. Mere 

 ,oe that the 5 is transposed ; it is taken from the 

 left side and put on the right with changed sign. The 

 new equation is therefore 4x = 3x-f 3. Again: if wo 

 now wish to remove the 3z from the right to the left, all 

 we have to do is to subtract 3z from both sides; tlio 

 result is the equation 4x 3u; = 3. Here you see that 

 the !!.c is transposed : it is taken from the right and put 

 on the left with changed sign. The new equation is 

 therefore x = 3. We have thus actually solved the equa- 

 tion 4x 6 = 3x 2 ; the x in tliis equation, at the 

 outset unknown in value, is now found out to stand 

 for 3. If you put this 3 for the x in the proposed equa- 

 tion, it becomes the numerical equation 12 6 = 9 2 ; 

 which you see is true, each side being 7. 



Clearing Fractions. When a fraction occurs in an 

 equation, wo may clear the equation of the fraction by 

 multiplying both sides of it by the denominator of the 

 fraction. You know from common arithmetic, that if 

 you multiply a fraction by its denominator, the result 



24 2 



is simply the numerator : thus, t x 3 = 2 ; - X 5=4; - X 



80 t 



7 = 2, and so on. In like manner, ? X 3 = x ; 



S 



X 5 



2a 



= 4x ; X 7 = -a, <tc. If, therefore, we have such an 



equation as 2x + * + 2 = 22 x it will become cleared 



3 



of fractions by multiplying both sides by 3 : the result 

 of this multiplication is the new equation 6x -f- x -f- G = 

 00 3x, which is free from fractions. To complete the 

 solution of the equation, that is, to find the value of r, 

 we must, by transposition, bring all the unknown quan- 

 tities to one side of the equation, and all the .known 

 quantities to the other : we shall thus have Ox + x -f- 

 3z = G6 6 ; that is, 10x = 00 : consequently, dividing 

 each side by 10, there results finally x = : and if this 

 bo put for x in the original equation, you will find the 

 two sides of it to be numerically equal, for each side 

 will bo 36. 



To solve a simple equation containing only one unknown 

 quantity, 



RULE I. If there be a fraction in the equation, clear 

 it away by multiplying both sides by the denominator of 

 the fraction. 



2. If known and unknown quantities are linb-1 to- 

 gether, separate them by transposition ; so that all the 

 unknown quantities may appear on one side of the equa- 

 tion, and all the known quantities on the other. 



3. Collect the terms on each side into one sum, so 

 that there may bo only a single unknown term on one 

 side of the equation, and a single known term on the 

 other. 



4. Lastly, divide each side by the coefficient or mul- 

 tiplier of x, the unknown quantity, the result will I 

 alone on one side, and its value or interpretation on the 

 other side of the sign of equality. 



EXAMPLE 1. Given the equation 5x 8 = 3^ + 2 to 

 find the value of x. As it is usual to choose tli< 

 hand side of the equation for the unknown quanta 

 and the other side for the known numbers, wo shall 

 traiutpuse the known number 8, and thus get the equa- 

 tion 5j; 3.C + 2 + 8 ; then transposing the unknown 

 quantity 3r, wo shall obtain the equation Six 3.t 2 -j- 8, 



