ON EQUATIONS. - PRELIMINARY REMARKS. 134. AN equation, in the most general acceptation of the term, signifies two alge braic expressions which are equal to each other, and are connected by the sign = Thus, a x = b, c x2 + dx = e, c x3 + g x2= h x + k, m x 1 + n x 3 + p x2 +qx+r=o, are equations. 4 The two quantities separated by the sign are called the members of the equation, the quantity to the left of the sign is called the first member, the quantity to the right the second member. The quantities separated by the signs + and are called the terms of the equation. 135. Equations are usually composed of certain quantities which are known and given, and others which are unknown. The known quantities are in general represented either by numbers, or by the first letters in the alphabet, a, b, c, &c.; the unknown quantities by the last letters, s, t, x, y, z, &c. 136. Equations are of different kinds. 1o. An equation may be such, that one of the members is a repetition of the other, as, 2 x − 5 = 2 x .5. 2o. One member may be merely the result of certain operations indicated in the other member, as, 5x+16= 10 x — 5 — (5 x −21), (x+y)(x − y) - 3 x * — ყ ვ = x2 — y2, x y 3o. All the quantities in each member may be known and given, as, 25 = 10 + 15, a+b=c- d, in which, if we substitute for a, b, c, d, the known quantities which they represent, the equality subsisting between the two members will be self-evident. = x2+ xy + y2. In each of the above cases the equation is called an identical equation. 4°. Finally, the equation may contain both known and unknown quantities, and be such, that the equality subsisting between the two members cannot be made manifest, until we substitute for the unknown quantity or quantities certain other numbers, the value of which depends upon the known numbers which enter into the equation. The discovery of these unknown numbers constitutes what is called the solution of the equation. The word equation, when used without any qualification, is always understood to signify an equation of this last species; and these alone are the objects of our present investigations. x+4=7 is an equation properly so called, for it contains an unknown quantity x, combined with other quantities which are known and given, and the equality subsisting between the two members of the equation cannot be made manifest, until we find a value for x, such, that when added to 4, the result will be equal to 7. This condition will be satisfied, if we make x = 3, and this value of a being determined, the equation is solved. The value of the unknown quantity thus discovered is called the root of the equation SIMPLE EQUATIONS. 137. Equations are divided into degrees according to the highest power of the unknown quantity which they contain. Those which involve the simple power only of the unknown quantity, are called simple equations, or equations of the first degree; those into which the square of the unknown quantity enters, are called quadratic equations, or, equations of the second degree; so we have cubic equations, or, equations of the third degree; biquadratic equations, or, equations of the fourth degree; equations of the fifth, sixth, nth degree. Thus, 197 138. Numerical equations are those which contain particular numbers only, in addition to the unknown quantity. Thus, x3 + 5 x2 = 3 x + 17, is a numerical equation. 3 3 139. Literal equations are those in which the known quantities are represented by letters only, or by both letters and numbers. Thus, x3+p x2 + q x = r2, x * — 3 p x 3 + 5 q x2 + 1 x = 5 are literal equations. 2 3 2 140. Let us now pass on to consider the solution of equations, it being understood, that, to solve an equation, is to find the value of the unknown quantity, or to find a number which, when substituted for the unknown quantity in the equation, renders the first member identical with the second. The difficulty of solving equations depends upon the degree of the equations, and the number of unknown quantities. We first consider the most simple case. ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 141. The various operations which we perform upon equations, in order to arrive at the value of the unknown quantities, are founded upon the following principles : If to two equal quantities, the same quantity be added, the sums will be equal. If from two equal quantities, the same quantity be subtracted, the remainders will be equal. If two equal quantities be multiplied by the same quantity, the products will be equal. If two equal quantities be divided by the same quantity, the quotients will be equal. These principles, when applied to the two equal quantities which constitute the two members of every equation, will enable us to deduce from them new equations, which are all satisfied by the same value of the unknown quantity, and which will lead us to discover the value of that unknown quantity. 142. The unknown quantity may be combined with the known quantities in the given equation, by the operations of addition, subtraction, multiplication, and division. We shall consider these different cases in succession. L. Let it be required to solve the equation, x + a = b If, from the two equal quantities x + a and b, we subtract the same quantity a, the remainders will be equal, and we shall have, x + a — a = ba or, xb-a, the value of x required. So also, in the equation, x+6=24 Subtracting 6 from each of the equal quantities x + 6 and 24, the result is, x = 24-6 = 18, II. Let the equation be, the value of x required. x—a—b If, to the two equal quantities x —a and b, the same quantity a be added, the sums will be equal, then we have, x—a+ab+ a or, So also in the equation, ≈ = b+a the value of x required. X- 6 24 Adding 6 to each of these equal quantities, the result is, x = 24 +6 = 30,. the value of x required. It follows from (I.) and (II.) that, We may transpose any term of an equation from one member to the other, by changing the sign of that term. We may change the signs of every term in each member of the equation, without altering the value of the expression. This is, in fact, the same thing af transposing every term in each member of the equation. If the same quantity appear in each member of the equation affected with the same sign, it may be suppressed. Dividing each of these equals by 6, the result is, x= 4, the value of x required. From this it follows, that, When one member of an equation contains the unknown quantity alone, affected with a coefficient, and the other member contains known quantities only, the value of the unknown quantity is found by dividing each member of the eque tion by the coefficient of the unknown quantity. Multiplying each of these equals by a, the result is, xab, the value of a required. So also in the equation, =24 6 Multiplying each of these equals by 6, the result is, = 144 From this it follows, that, When one member of the equation contains the unknown quantity alone, divided by a known quantity, and the other member contains known quantities only, the value of the unknown quantity is found by multiplying each member of the equation by the quantity which is the divisor of the unknown quantity. V. Let the equation be, a en r ben ах In order to solve this equation, we must clear it of fractions; to effect this, reduce the fractions to equivalent ones, having a common denominator (Art. 52), the equation becomes, 53 12 10 x 24 - с = Hence, it appears that, c = Multiply these equal quantities by the same quantity ben, or, which is evidently the same thing, suppress the denominator ben in each of the fractions, and multiply the integral term by ben, the result is, aenz - bce n = bdnx — bem, an equation clear of fractions. So also in the equation, 2 x 3 Reducing the fractions to a common denominator, 40 x 45 = 11+ 60 dx 4 x Multiplying both members of the equation by 60, the result is, 40 x bdn x ben 32 x 24 If the denominators have common factors, we can simplify the above operation by reducing them to their least common denominator, which is done (see arithmetic) by finding the least common multiple of the denominators. Thus, in the equation, m n 3 x 4 = 11 + 1/1/ 5 13 = bem ben 7 8 12 x 60 The least common multiple of the numbers 12, 3, 8, 6, is 24, which is therefore the least common denominator of the above fractions, and the equation will become, 21 24 13 = Multiplying both members of the equation by 24, the result is, 10 x 32 x 312 21 — - 52 x, an equation clear of fractions. In order to clear an equation of fractions, reduce the fractions to a common Jenomizarɔr, multiply each integral term by this common denominator, and then suppress it. 143. From what has been said above, we deduce the following general RULE FOR THE SOLUTION OF A SIMPLE EQUATION CONTAINING ONE UNKNOWN QUANTITY. 1o. Clear the equation of fractions, and perform in both members all the alge braic operations indicated. 2o. Transpose all the terms containing the unknown quantity to one member of the equation, and all the terms containing known quantities only to the other member of the equation, and reduce each member to its most simple form. 3o. We thus obtain an equation, one member of which contains the unknown quantity alone, affected with a coefficient, and the other member contains known quantities only; the value of the unknown quantity will be found by dividing the member composed of the known quantities by the coefficient of the unknown quantity. The terms containing the unknown quantity are usually collected in the first member of the equation. |