Exercises.

Transpose the unknown terms to the first member, and the known terms to the second, in the following:

107. The solution of an equation is the operation of finding such a value for the unknown quantity as will satisfy the equation; that is, such a value as, being substituted for the unknown quantity, will render the two members equal. This is called a root of the equation.

A root of an equation is said to be verified, when, being substituted for the unknown quantity in the given equation, the two members are found equal to each other.

Take the equation

Clearing of fractions (§ 104), and performing the operations indicated, we have

12x-32=4x-8+24.

Transposing all the unknown terms to the first member, and the known terms to the second (§ 106), we have

Hence 6 satisfies the equation, and therefore is a root.

108. By processes similar to the above, all equations of the first degree, containing but one unknown quantity, may be solved. Hence the rule:

Clear the equation of fractions, and perform all the indicated operations.

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

Reduce all the terms in the first member to a single term, one factor of which shall be the unknown quantity, and the other factor will be the algebraic sum of its coefficients.

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

a result which may be verified by substituting it for x in the given

equation.

2. Solve the equation

(3 a − x)(a−b)+2ax=4b (x+a).

Performing the indicated operations, we have 3a2-ax-3 ab + bx + 2 ax = 4 bx +4 ab.

3. Given 3x-2+24=31, to find x.

4. Given x+18=3x-5, to find x.

5. Given 6-2x+10=20-3x-2, to find x.

Ans. x =

= 3.

Ans. x=11.

18.

a + x

8

8 ax b

7

---

3

α с

α х

2

3b

=

2

a2

с

13 3

In the following equations, what is the numerical value of x, when a = 1, b = 2, c = 3,' d = 4, and ƒ=6?

109. A problem is a question proposed, requiring a solution. 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.

The statement consists in expressing algebraically the relation between the known and the required quantities.

The solution consists in finding the values of the unknown quantities in terms of those which are known.

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 problems is best learned by practical examples.

(1) What number is that to which if 5 be added the sum will be equal to 9?

This is the statement of the problem.

To find the value of x, transpose 5 to the second member.