Examples, wherein the foregoing Rules appear. EXAMPLE I. What is the value of x, in the equation 10436 = 16. 36 12-x By subtracting 10 from each fide of the equation, we have 12-x =6, both fides of which divided by 6, the quotient 6 is=1, this multiplied by 12-x, gives 6=12-X, whence by tranfpofing x and 6, we have x=12-6, or x=6. EXAMPLE 2. What is the value of x in the equation ax2 + ac2 =ax+b2. Here multiplying by a+x, there comes a+x out ax2+ac2=ax+b2xa+x, or ax2+ac2=a2x+ab2+ax2+ b2x, which tranfpofed and ordered according to the foregoing -ab2+ac2 rules, is a2x+6x=—ab2+ac2, wherefore, x= a2+62 fo that if a 1, b=2, c=3, then will x 4+9. 1+4 EXAMPLE 3. What is the value of x in the equation ↓·a2+x2— A b++. Both fides of this equation being failed to the fourth power, we have a++2a2x2+x^=b++xa, which by tranfpofition, &c, becomes za2x2—ba—aa, which b4a4 242 242 therefore x= divided by 2a2, becomes a2= EXAMPLE 4. What is the value of x in the following ̈equation: x=√22+x+c. In this equation x+c ++, which fquared gives a2+2cx+c2=c2+ x/b2+x2, or x2+2¢x=x√/b2+x2; dividing this by x, it quotes x+2c=√6+; this fquared gives x2+4x+482= +, and by tranfpofition it becomes 4cx=-462; and dividing by 4 we have x= 4c To exterminate an unknown Quantity out of feveral Equations; or, to reduce two or more Equations to a fingle one. RULE. If the quantity to be exterminated, has but one dimenfion in the equation, find the value of it in two equations, and put those values equal to each other; or having found the value in one equation, fubstitute it in the room of the quantity in the other equations. Proceed in the fame manner with every unknown quantity. But if the quantity to be exterminated be of feveral dimenfions, find the value of its highest power in two equations. Then if the coefficients are not the fame, multiply the lefs quantity, fo that it may become equal to the greater. Put these values equal to each other, and there will arife a new equation, with a lefs power of the unknown quantities: and the operation must be repeated till the quantity be exterminated. Examples. EXAMPLE I. What is the value of x and y in these twe equations, 7x-5y=28 and 3x+4y=55, by transpofing 28 in the first equation, and 57, we have 7-285, therefore 7x-28 S the value of y is In In the fecond equation by proceeding in the fame manner, viz.By tranfpofing 3x, the value of y is found to be 55-3; therefore, these two quantities being put equal 4 to each other, we have the equation 7-28_55—3x. In 5 4 this equation only a is concerned. Multiply this equation by 20, which is the production of 4 and 5; or, which is the fame thing, multiply the numerators and the denominators cross ways, and there will arife the equation 28x-112=275—15x, 387 which by tranfpofition becomes 43x=387, or x=- 43 therefore, x=9; therefore, 9 being substituted in either of the given equations instead of x, the value of y will be found. Thus, in the firft equation if 9 be fubftituted for x, it will و be 63-5y=28, which tranfpofed is 63-28 =y, or 7=y. EXAMPLE 2. Required the value of x, y, and z, in the three following questions: x+100=y+z; y+100=2x+2%; ≈+100=3x+3y, by tranfpofing 100 in the first equation, xy+x-100 arifes, which value fubftituted in the other two equations, instead of x, we have the two following: y+100 (2y+2x-200+2x)=2y+4x-200 +100 (3y+3x-300+3y)=6y+3%-300, then by tranfpofing, and 42—200, in the first of these two equations we have 300-42=y, which fubstituted for the y in the last equation, is z+100=1800—24%+32-300, that is z+100 =1500-212; wherefore 22x=1400, or z= =63 ጹ፡ 1400 22 therefore, y=300-4%-451, and xy+x-100=97r. Of Of the Nature and Compofition of Equations, containing different Dimenfions of the fame unknown Quantity. It often happens that the unknown quantity will be of feveral different dimenfions, then fuch equation is called a quadratic, a cubic, a biquadratic equation, &c. according as the dimenfion of the highest power is a fquare, cube, or biquadrate: in fuch equations we must discover the root or value of the unknown quantities. All equations are derived, (or may be confidered fo,) from those of a more fimple form. Thus, if x-bo, which is a fimple equation, be raised to the second power, there arises x2-2bx+b2=0, which is called a quadratic equation; if the former equation be raised to the third power, we have x3-—3b.x2+3b2x—b3=o, which is called a cubic equation, and fo on. It is but feldom that equations occur in this regular form, for the coefficients of the terms will generally be more or less than those produced by the involution of one quantity, as x-b, and therefore, a quadratic equation is a compound one, generally derived from xbxxc. A cubic equation is derived from xbxx-xx-d. A biquadratic equation from x-bxx-xx-xx-e, or from a quadratic fquared, &c. But, the letters b, c, d, &c. may have either affirmative or negative figns. In equations of this nature, as the whole is equal to nothing, it is obvious that fome or other of the factors must be equal to nothing. It is alfo evident that any fuch equation may be divided by its factors, till there remain only one factor; and as each of the inferior equations obtained by fuch divifion, muft ftill be equal to nothing, it must follow that each of these factors themselves are equal to nothing; therefore, b, c, d, e, &c. exhibit fo many different values of a with contrary figns; therefore every equation has as many roots roots as there are dimenfions of the unknown quantity in its higheft power. And where b, c, d, e, &c. are found negative, x is affirmative; and where any of thefe are affirmative, is negative. By multiplying the factors or roots together, under different figns, it is obfervable that when b, c, d,' &c. are all negative, or, which is the fame thing, when all the values of x are affirmative, the figns in the equation are +, and alternately. But when there is a negative root, one affirmative quantity will follow another; therefore, there will be as many affirmative roots in the equation as there are changes of the figns from + to, and from to +, and all the reft will be negative. What is here delivered, concerns only poffible roots. An impoffible root is when b, c, d, &c. denote the fquare or any other even root of a negative quantity; an equation derived from fuch roots is an impoffible or imaginary one: if there be one poffible root, the equation will admit of one poffible answer. In the multiplication of the roots of fuch equations, the coefficient of the second term is the fum of all the roots with contrary fines; the coefficient in the third term is equal to the fum of the rectangles of those roots; or, of all the products that can poffiby arife by combining them two and two; the coefficient of the fourth term is equal to the fum of all the products that can poffibly arife by the combination of them three and three, &c. and the laft term is always equal to the product of all the roots with contrary figns. The Refolution of Quadratic Equations. 20 If it be a pure quadratic, as 2=b2, or 2-b2-0, it is produced from the rectangle of a-b and x + b, and therefore has one affirmative, and one negative root, and the affirmative root is equal in number to the negative. The root in this -cafe is fouud by extracting the fquare root of the number dented by b. Thus, if 2+=576, then +576=+24. = All |