or EXAMPLE 2. Solve for x: 3-x-2(x-1)(x+2) = (x-3)(5—2x). Multiply out the parentheses: 3-x-2(x+x-2) = 11x- 2x2-15 -- 3 X 2x2 -2x+4 = 11x 2 x2 15. Transpose the terms involving the unknown quantities: -x-2x2-2x-11x+2x2-3-4-15 = Combine first the terms in the first member of the equation, them Similarly, the terms in the second member combined give, FORMULAE FOR THE SOLUTION OF AN EQUATION OF THE FIRST DEGREE IN ONE UNKNOWN QUANTITY 174. Every equation of the first degree in one unknown quantity can, as has been seen, be reduced by addition, subtraction, and multiplication to the form, ax = b, To arrive at this result, remove where a and b are known numbers. fractions, and render the equation integral throughout by multiplying both members by the least common denominator. Transpose all the terms in x to the first member, and all the known terms to the second. Then combine all the terms in x into one term, and, similarly, all the known terms into a single term. The equation having been reduced to the form, two principal cases can arise: either a, the coefficient of x, is different from zero or it is equal to zero. 175. When a is Different from Zero.-If a is different from zero, divide both members of the equation by a and form equation (2), equivalent to equation (1): Since a is different from zero, equation (1) has a determinate root, and this root is given by formula (2). 176. When a is Equal to Zero.-In case a is zero, it is no longer possible to divide both members of equation (1) by a. It is accordingly necessary to study this equation more minutely. Two cases can arise: at the same time that a is zero, b can be different from zero or equal to zero. 1. When a = 0, but b 0. In this case, no number substituted for x can satisfy equation (1), because the product of any number whatever by a, that is to say, by zero, is equal to zero, and consequently is different from b. The equation is therefore impossible. Suppose that instead of a's being zero, a is very small; then the equation, ax = b, and accordingly will have a determinate root. If b remains fixed and the number a decreases indefinitely, and approaches zero, the root b will increase indefinitely, and in case a is equal to zero, the equa a tion is said to have an infinite* indeterminate root (873, 2). 2. When a = 0, and b = 0. Then any number put in place of x will satisfy the equation, because the product of any number whatever by 0 is equal to zero. The value of x, O, is then indeterminate (₹73, 1). 177. Numerical Applications.-Consider the equations: Consequently equation (1) has a determinate root 12.59259..... * An infinite number is one which is larger than any number one can choose. |
