Identity is therefore satisfied by any value whatever of the unknown quantity; whereas in "Equations" the unknown quantities have particular values, which alone, and none other, will permit the expressed equality to subsist. To find these values is to "Solve" the Equations, and forms an important part of the business of Algebra. These values are sometimes called the "Roots" of the Equations, and are said to satisfy them. Thus, if 22-6 be the Equation, x=3, and can be nothing else; and x=3 is called its solution. Again, if x=4, we know that x=2, or -2; and 2, −2, are called the Roots of the equation x2=4. 185. When an equation is cleared of fractions and surds, if it contain the first power only of an unknown quantity, it is called a Simple Equation, or an equation of one dimension; if the square of the unknown quantity be in any term, (and there be no higher power,) it is called a Quadratic, or an equation of two dimensions; if the Cube of the unknown quantity appear, (and no higher power,) it is called a Cubic Equation; if the fourth power, a Biquadratic; and in general, if the index of the highest power of the unknown quantity be n, it is called an equation of n dimensions. SIMPLE EQUATIONS. 186. RULE. I. In any equation quantities may be transposed from one side to the other, if their signs be changed, and the two sides will still be equal. For let +10=15; then by subtracting 10 from each side, (Art. 80), x+10-10-15-10, or a=15-10. Let -4-6; by adding 4 to each side, (Art. 79), If x-a+b=y; adding a-b to each side, x-a+b+a-b-y+a-b; or x=y+a-b. 187. COR. Hence, if the signs of all the terms on each side be changed, the two sides will still be equal. 188. RULE II. If every term on each side be multiplied by the same quantity, the results will be equal (Art. 81). 189. COR. An equation may be cleared of fractions, by multiplying every term successively by the denominators of those fractions. An equation may be cleared of fractions at once, by multiplying both sides by the product of all the denominators, or by any quantity which is a multiple of them all. multiplying by 2×3×4, 3×4×~+2×4×v + 2×3×x = 2×3×4×13, or 12x+8x+6x=312; that is, 26x = 312. If the Least Common Multiple of the denominators be made use of, the equation will be in the lowest terms. Thus, if each side of the last equation be multiplied by 12, which is the Least Com. Mult. of 2, 3, and 4, the equation will become 190. RULE III. If each side of an equation be divided by the same quantity, the results will be equal. (Art. 82.) Let 17=136; then = 191. 136 RULE IV. If each side of an equation be raised to the same power, the results will be equal. (Art. 81.) Let -9; then = 9×9=81. = Also, if the same root be extracted on both sides, the results will be equal. Let = 81; then 9 (Art. 143). x= = An equation may be cleared of a surd by transposing the terms so that the surd shall form one side, and the rational quantities the other, and then raising both sides to that power which will rationalize the surd. Thus, if √a+x-b-c, by transposition Ja+x=b+c, and a+x=(b+c)2. (Art. 81.) If the equation contain two surds, connected by + or, then the same operation must be repeated for the second surd. an equation in which the surds do not appear. 193. A "simple equation" can have only one solution; that is, there can be but one value of the unknown quantity which satisfies it. For every "simple equation" with respect to the unknown quantity x can be reduced to the form ax+b=0. Now, if possible, let there be two values of x which satisfy this equation, viz. a and ß; then aa+b=0, and aẞ+b=0; .. subtracting, aa-aẞ=0, or a(a-3)=0. But a cannot be equal to 0, for then the proposed equation would be no equation at all with respect to r, therefore a-ẞ=0, or a=ß; that is, a and ẞ cannot be different values; or there is only one value of x which satisfies the equation. If however it be known that a is not ß, i. e. that the proposed equation has two different roots, the equation u(a-3)=0 cannot subsist unless a=0, and then will also =0; i.e. the equality ax+b=0 ceases to be an equation, and becomes an identity, the coefficient of r and the other term becoming separately =0. 194. To find the value of the unknown quantity in a simple equation. Let the equation first be cleared of fractions and surds* (Arts. 189, 192), then transpose all the terms which involve the unknown • It should be borne in mind that this is required to be done only when the unknown quantity is found in a fraction or surd. Thus it will not be necessary in such equations as the following: 4 6 quantity to one side of the equation, and the known quantities to the other (Art. 186); divide both sides by the coefficient, or sum of the coefficients, of the unknown quantity (Art. 190), and the value required is obtained. Ex. 1. To find the value of x in the equation 3x − 5 = 23 − x. |