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The equationx+7=8x-12 has two terms in each of its members, namely,

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Although I have taken at random, and to serve for an example merely, the equation + 78 x 12, it is to be considered, as also every other of which I shall speak hereafter, as derived from a problem, of which we can always find the enunciation by translating the proposed equation into common language. This under consideration becomes,

To find a number x such, that by adding 7 to 3x, the sum shall be equal to 8 times x minus 12.

Also the equation ax+bc-cxac-b x, in which the letters a, b, c, are considered as representing known quantities, answers to the following question;

To find a number x such, that multiplying it by a given number a, and adding the product of two given numbers b and c, and subtracting from this sum the product of a given number c by the number x, we shall have a result equal to the product of the numbers a and c, diminished by that of the numbers h and x.

It is by exercising one's self frequently in translating questions from ordinary language into that of algebra, and from algebra into ordinary language, that one becomes acquainted with this science, the difficulty of which consists almost entirely in the perfect understanding of the signs and the manner of using them.

To deduce from an equation the value of the unknown quantity, or to obtain this unknown quantity by itself in one member and all the known quantities in the other, is called resolving the equation.

As the different questions, which are solved by algebra, lead to equations more or less compounded, it is usual to divide them into several kinds of degrees. I shall begin with equations of the first degree. Under this denomination are included those equations in which the unknown quantities are neither multiplied by themselves nor into each other.

Of the resolution of Equations of the First Degree, having but one unknown quantity.

9. WE have already seen that to resolve an equation is to arrive at an expression, in which the unknown quantity alone in

one member is equal to known quantities combined together by operations which are easily performed. It follows then, that in order to bring an equation to this state, it is necessary to free the unknown quantity from known quantities with which it is connected. Now the unknown quantity may be united to known quantities in three ways;

1. By addition and subtraction, as in the equations,

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2. By addition, subtraction, and multiplication, as in the equations,

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3. Lastly, by addition, subtraction, multiplication, and divis ion, as in the equations,

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The unknown quantity is freed from additions and subtractions, where it is connected with known quantities, by collecting together into one member all the terms in which it is found; and for this purpose it is necessary to know how to transpose a term from one member to the other.

10. For example, in the equation

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5124 x,

it is necessary to transpose 4 x from the second member to the first, and the term 5 from the first member to the second. In order to this, it is obvious, that by cancelling + 4x in the second member, we diminish it by the quantity 4 x, and we must make the same subtraction from the first member, to preserve the equality of the two members; we write then 4x in the first member, which becomes 7 x 4 x and we have

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5
4 x 12.

To cancel 5 in the first member, is to omit the subtraction of 5 units, or in other words, to augment this member by 5 units; to preserve the equality then we must increase the second member by 5 units, or write + 5 in this member, which will make it 125; we have then

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By performing the operations indicated there will result the equation

3 x 17.

=

From this mode of reasoning, which may be applied to any example whatever, it is evident, that to cancel in a member a term affected with the sign +, which of course augments this member, it is necessary to subtract the term from the other member, or to write it with the sign; that on the contrary when the term to be effaced has the sign minus, as it diminishes the member to which it belongs, it is necessary to augment the other member by the same term, or to write it with the sign +; whence we obtain this general rule;

To transpose any term whatever of an equation from one member to the other, it is necessary to efface it in the member where it is found, and to write it in the other with the contrary sign.

To put this rule in practice, we must bear in mind that the first term of each member, when it is preceded by no sign, is supposed to have the sign plus. Thus, in transposing the term ex of the literal equation a x - b = c x + d from the second member to the first, we have

ax - b— cx = d;

transposing alsob from the first member to the second, it becomes

а x- cx = d + b.

11. By means of this rule, we can unite together in one of the members all the terms containing the unknown quantity, and in the other all the known quantities; and under this form the member, in which the unknown quantity is found, may always be decomposed into two factors, one of which shall contain only known quantities, and the other shall be the unknown quantity by itself.

This process suggests itself immediately, whenever the proposed equation is numerical and contains no fractions, because then all the terms involving the unknown quantity may be reduced to one. If we have, for example,

10x+7x 2 x = 25+ 7,

by performing the operations indicated in each member, we shall

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and 15x is resolved into two factors 15 and x; we have then

the unknown factor x by dividing the number 32, which is equal to the product 15 x by the given factor 15, thus,

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This resolution is effected in like manner in the literal equations of the form

ax = bc;

because the term a signifies the product of a by x; we hence conclude, that

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which contains three terms involving the unknown quantity. Since a x, b x, cx, represent the products respectively of x by the quantities a, b, and c, the expression axbx + cx translated into ordinary language is rendered as follows,

From x taken first, so many times as there are units in a, subtract so many times x as there are units in b, and add to the result the same quantity x, taken so many times as there are units in c.

It follows then on the whole, that the unknown quantity x is taken so many times as there are units in the difference of the numbers a and b, augmented by the number c, that is to say, so many times as is denoted by the number a b+c; the two factors of the first member are therefore a have then

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b+c and

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we

From this reasoning which may be applied to every other example, it is evident, that after collecting together into one member the different terms containing the unknown quantity, the factor, by which the unknown quantity is multiplied, is composed of all those quantities by which it is separately multiplied, arranged with their proper signs, and the unknown quantity is found by dividing all the terms of the known member by the factor which is thus obtained. According to this rule, the equation ax-3x= b c gives

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for it is necessary to observe that the letter, taken singly, must be regarded as multiplied by one. It is besides manifest, that in xax, the unknown quantity a is contained once more than in a x, and is consequently multiplied by 1 + a.

12. It is evident that if there be a factor, which is common to all the terms of an equation, it may be dropped without destroying the equality of the two expressions, since it is merely dividing by the same number all the parts of the two quantities, which are by supposition equal to each other.

Let there be, for example, the equation

6 abx9bcd12b dx + 15 abc.

I observe in the first place, that the numbers 6, 9, 12 and 15 are divisible by 3, and by suppressing this factor, I merely take a third part of all the quantities which compose the equation.

I have after this reduction,

2 a b x 3 b c d

= 4 bdx + 5 abc.

I observe, moreover, that the letter b, combined in each term as a multiplier, is a factor common to all the terms; by cancelling it the equation becomes

2 ax 3 cd = 4 dx + 5 ac.

Applying the rules given in articles 10 and 11, I deduce successively

sors.

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13. I now proceed to equations, the terms of which have diviThese may be solved by the preceding rules whenever the unknown quantity does not enter into the denominators; but it is often more simple to reduce all the terms to the same denominator which may then be cancelled.

Let there be, for example, the equation

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Arithmetic furnishes rules for reducing fractions to the same denominator, and for converting whole numbers into fractions of a given kind. (Arith. 79, 69.) Let all the terms of the proposed equation be transformed by these rules into fractions of the same denominator, beginning with the fractions, which are

2x 4x 5x

3' 5' 7

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