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Clearing of fractions (Art. 83), and performing the operations indicated, we have

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Transposing all the unknown terms to the first member, and the known terms to the second member (Art. 84), we have,

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Reducing the terms in the two members, we have

5 – 15.

Dividing both members by the coefficient of x, we have,

x = 3.

In the same way, all equations of the first degrec, containing but one unknown quantity, may be solved. Hence, the following

RULE.

I. Clear the equation of fractions, and perform al. the indicated operations.

II. Transpose all of the unknown terms to the first member, and all the known terms to the second member.

III. Reduce all the terms in the first member to a single term, one factor of which is the unknown quantity; the other factor will be the algebraic sum of its coeffi cicnts.

IV Divide both members by the coefficient of the un known quantity: the second member will be the value of the unknown quantity.

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transposing and reducing, - (a3b+ ab3)x = —

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a2;

(a2 + b2);

=

a3b + ab3 ab

Ans. x 2.

11

Ans. x 12.

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87. A PROBLEM is a question proposed, requiring a volution.

88. The SOLUTION of a problem is the operation of finding a quantity, or quantities, that will satisfy the given conditions.

The solution of a problem consists of two parts; the

statement, and the solution of the equation or equa tions of the problem.

The STATEMENT consists in translating the conditions of the problem into algebraic language, the resulting equations being called the equations of the problem. The solution of the equations is made by the general rules for solving equations.

The statement is made by representing the unknown quantities of the problem by some of the final letters of the alphabet, and then operating upon these so as to comply with the conditions of the problem. The method of stating a problem will be best learned from practical examples.

1. What number is that to which if its fifth part be added, the sum will be equal to 24?

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2. The sum of two numbers is 30, and their differ

ence 6. What are the numbers?

Let x denote one number.

Then will 30 x, denote the other.

From the conditions,

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