Questions producing Equations of the First Degree, involving but one Unknown Quantity.

94. It has already been observed (Art. 81), that the solution of a problem by Algebra, consists of two distinct parts.

1st. The statement; and

2d. The solution of the equation.

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

This part cannot, like the second, be subjected to any welldefined 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 equation 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 are enabled to discover the equation by applying the following

RULE.

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

QUESTIONS.

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

х.

one third of it

one fourth of it

+45 = 448.

And by the conditions, 2+3+

Now, by subtracting 45 from both members,

By making the terms of the equation entire, we obtain

Let us see if this value will verify the equation of the prob

lem. We have

372 +3 +372 +372

2

+45=186 +124 +93 + 45 = 448.

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

by 16.

Let the required number be represented by x. Then

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

Suppose the cask to have held a gallons.

+ 21 = what leaked out, and what was drawn.

Hence, + 21 =
3

or

or

or

by changing the signs of both members, which does not destroy their equality.

4. A fish was caught whose tail weighed 9lb.; his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail together: what was the weight of

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

and

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.

General Solution.

the whole number of working and idle days,
the amount received for each day he worked,
the amount paid for his board, for each idle day,
the balance due, or the result of the account,
the number of working days,
the number of idle days.

6. A fox, pursued by a greyhound, has a start of 60 leaps. He makes 9 leaps while the greyhound makes but 6; but 3 leaps of the greyhound are equivalent to 7 of the fox. How many leaps must the greyhound make to overtake the fox ?

From the enunciation, it is evident that the distance to be passed over by the greyhound, is equal to the 60 leaps of the fox, plus the distance which the fox runs after the greyhound starts in pursuit.

Let x = the number of leaps made by the greyhound from the time of starting till he overtakes the fox.

Now, since the fox makes 9 leaps while the greyhound makes

3

6, the fox will make 11, or leaps while the greyhound makes 1; and, therefore, while the greyhound makes a leaps, the