Whence, by dividing both members of the equation by 30,

x = 11.1.

If we substitute this value of x, for x, in the given equation, it will verify it, that is, make the two members equal to each other.

Find the value of x in each of the following

Problems giving rise to Equations of the First Degree, involv ing but une Unknown Quantity.

81. The solution of a problem, by means of algebra, consists of two distinct parts

1st. The statement of the problem; and

2d. The solution of the equation.

We have already explained the methods of solving the equa tion; and it only remains to point out the best manner of making the statement.

The statement of a problem is the operation of expressing, algebraically, the relations between the known and unknown quantities which enter it.

This part cannot, like the second, be subjected to any well defined rule. Sometimes the enunciation of the problem furnishes the equation immediately; and sometimes it is necessary to discover, from the enunciation, new conditions from which an equa tion may be formed.

The conditions enunciated are called explicit conditions, and those which are deduced from them, implicit conditions.

In almost all cases, however, we equation by applying the following

are enabled to discover the

RULE.

Denote the unknown quantity by one of the final letters of the alphabet, and then indicate, by means of algebraic signs, the same operations on the known and unknown quantities, as would be necessary to verify the value of the unknown quantity, were such

value known.

PROBLEMS.

1. Find a number such, that the sum of one half, one third and one fourth of it, augmented by 45, shall be equal to 448 Let the required number be denoted by

Then, one half of it will be denoted by

X.

x

+ ૨ ૦૭/ ૨ ૩

6x+4x+3x

=

4836;

13x= 4836;

x = 372.

Let us see if this value will verify the equation. We have,

2. What number is that whose third part exceeds its fourth

3. Out of a cask of wine which had leaked away a third part, 21 gallons were afterward drawn, and the cask was then half full how much did it hold?

Suppose the cask to have held x gallons.

+ 21 =

;

4. A fish was caught whose tai weighed 916.; his head weighed as much as his tail and half his body; his body weighed as much as his head and tail together: what was the weight of the fish? denote the weight of the body;

Let

then

2r

9x

will denote weight of the head;

and since the body weighed as much as both head and tail,

5. A person engaged a workman for 48 days. For each day that he labored he received 24 cents, and for each day that he was idle, he paid 12 cents for his board. At the end of the 48 days the account was settled, when the laborer received 504 cents. Required the number of working days, and the number of days he was idle.

If these two numbers were known, by multiplying them respectively by 24 and 12, then subtracting the last product from the first, the result would be 504. Let us indicate these operations by means of algebraic signs.

Let

then

48 -x

24 X. X

=

=3

denote the number of working days;
will denote the number of idle days;
the amount earned, and

12 (48) the amount paid for his board. Then, from the conditions,