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To find a number which being added to itself, shall give a sum equal to 30. Were it required to solve this question, we should first express it in algebraic language, which would give the equation ac-Ha!-30. By adding a to itself, we have 2a – 30. And by dividing by 2, we obtain ac-15. Hence we see that the solution of a question by algebra consists of two distinct parts. 1st. To express algebraically the relation between the known and unknown quantities. 2nd. To find a value for the unknown quantity, in terms of those which are known. This latter part is called the solution of the equation.
The given or known parts of a question, are represented either by numbers or by the first legiers of the alphabet, a, b, c, &c. The unknown or required parts are represented by the final letters, r, !), 2, &c.
Find a number which, being added to twice itself, the sum shall be equal to 24.
Quest.—62. How may you regard every equation? What question does the equation z+z=30 state? Of how many parts does the solution of a question by algebra consist! Name them. What is the 2nd part called ! By what are the known parts of a question represented By what are the unknown parts represented
Statement. Let a represent the number. We shall then have a;+2a:=24.
This is the statement.
Solution. Having . . . . a-H23–24, we add . . . . . a 4-2a, which gives . . . 32–24; and dividing by 3, . ac-8.
63. An equation is said to be verified when the answer found, being substituted for the unknown quantity, proves the two members of the equation to be equal to each other.
Thus, in the last equation we found w=8. If we substitute this value for a in the equation
64. An equati, involving only the first power of the unknown quantity, is called an equation of the first degree.
are equations of the first degree. By considering the nature of an equation, we perceive that it must possess the three following properties:
QUEst.—63. When is an equation said to be verified ? 64. When an equation involves only the first power of the unknown quantity, what is it called 7 What are the three properties of every equation ?
1st. The two members are composed of quantities of the same kind: that is, dollars=dollars, pounds= pounds, &c.
2nd. The two members are equal to each other.
3rd. The two members must have the same sign.
65. An axiom is a self-evident truth. We may here state the following. 1. If equal quantities be added to both members of an equation, the equality of the members will not be destroyed. 2. If equal quantities be subtracted from both members of an equation, the equality will not be destroyed. 3. If both members of an equation be multiplied by the same number, the equality will not be destroyed. 4. If both members of an equation be divided by the same number, the equality will not be destroyed.
Transformation of Equations. 66. The transformation of an equation consists in changing its form without affecting the equality of its members.
The following transformations are of continual use in the resolution of equations.
67. When some of the terms of an equation are fractional, to reduce the equation to one in which the terms shall be entire.
1. Take the equation
Quest—65. What is an axiom Name the four axioms. 66. What is the transformation of an equation ? 67. What is the first transformation? What is the least common multiple of several numbers ? How do you find the least common multiple 2
First, reduce all the fractions to the same denominator, by the known rule; the equation then becomes
and since we can multiply both members by the same mumber without destroying the equality, we will multiply them by 72, which is the same as suppressing the denominator 72, in the fractional terms, and multiplying the entire term by 72; the equation them becomes
But this last equation can be obtained in a shorter way, by. finding the least common multiple of the denominators. The least common multiple of several numbers is the least number which they will separately divide without a remainder. When the numbers are small, it may at once be determined by inspection. The manner of finding the least common multiple is fully shown in Arithmetic $ 87. Take for example, the last equation
We see that 12 is the least common multiple of the denominators, and if we multiply all the terms of the equation by 12, and divide by the denominators, we obtain
68. Hence, to make the denominators disappear from an equation, we have the following
I. Find the least common multiple of all the denominators.
II. Multiply each of the entire terms by this multiple, and each of the fractional terms by the quotient of this multiple divided by the denominator of the term thus multiplied, and omit the denominators of the fractional terms.
1. Clear the equation of +++–4=3 of its denomi
nators. Ans. 7,452–140–105. 2. Clear the equation : + +-to-8 of its d - Q g-For-g=8 or its denomi* nators. Ans. 9a–H 62–2a:=432. - a o' ac a' - t t ---------- 3. Clear the equation 2 + 3 5. E 20 orie de nominators. Ans. 183-H122–4x-i-32=720. - ac Q? ac 4. Clear the equation +++ -ā-4 of its denominators. Ans. 14a:-H 102–35ac=280. 5. Clear the equation T-E-F-5- 15 of its denomi
nators Ans. 152–12a-H 10a:=900.
Quest.—68. Give the rule for clearing an equation of its denominators.