fluous. The four parts into which 45 is to be divided may be represented thus:

for if the first expression be increased by 2, the second dimin ished by 2, the third multiplied by 2, and the fourth divided by 2, the result in each case will be x. The sum of the four parts is 4x, which must equal 45.

Therefore the parts are 8, 12, 5, and 20.

Prob. 46. Divide the number a into four such parts that the first increased by m, the second diminished by m, the third multiplied by m, and the fourth divided by m, shall all be equal.

Ans.

ma

ma

α

m2a

-m; +m; (m+1)2; (m+1)2*

(m+1)2 (m+1)2

Prob. 47. A merchant maintained himself for three years at an expense of $500 a year, and each year augmented that part of his stock which was not thus expended by one third thereof. At the end of the third year his original stock was doubled. What was that stock?

Prob. 48. A merchant supported himself for three years at an expense of a dollars per year, and each year augmented that part of his stock which was not thus expended by one third thereof. At the end of the third year his original stock was doubled. What was that stock?

Prob. 49. A father, aged 54 years, has a son aged 9 years. In how many years will the age of the father be four times that of the son?

Prob. 50. The age of a father is represented by a, the age of his son by b. In how many years will the age of the father be n times that of the son?

Ans.

а-по N-1

CHAPTER IX.

EQUATIONS OF THE FIRST DEGREE CONTAINING MORE THAN ONE UNKNOWN QUANTITY.

145. If we have a single equation containing two unknown quantities, then for every value which we please to ascribe to one of the unknown quantities, we can determine the corresponding value of the other, and thus find as many pairs of values as we please which will satisfy the equation. Thus, let

If y=1, we find x=6; if y=2, we find x=4, and so on; and each of these pairs of values, 1 and 6, 2 and 4, etc., substituted in equation (1), will satisfy it.

Suppose that we have another equation of the same kind, as, for example,

5x+3y=19.

(2.)

We can also find as many pairs of values as we please which will satisfy this equation.

But suppose we are required to satisfy both equations with the same set of values for x and y; we shall find that there is only one value of x and one value of y. For, multiply equation (1) by 3, and equation (2) by 4, Axiom 3, and we have

Subtracting equation (3) from equation (4), Axiom 2, we have

whence

14x=28;
x=2.

(5.)

(6.)

Substituting this value of x in equation (1), we have

whence

4+4y=16;

y=3.

(7.)

(8.)

Thus we see that if both equations are to be satisfied, x must equal 2, and y must equal 3. Equations thus related are called simultaneous equations.

146. Simultaneous equations are those which must be satisfied by the same values of the unknown quantities.

When two or more simultaneous equations are given for solution, we must endeavor to deduce from them a single equation containing only one unknown quantity. We must therefore make one of the unknown quantities disappear, or, as it is termed, we must eliminate it.

147. Elimination is the operation of combining two or more. equations in such a manner as to cause one of the unknown quantities contained in them to disappear.

There are three principal methods of elimination: 1st, by addition or subtraction; 2d, by substitution; 3d, by comparison.

148. Elimination by Addition or Subtraction.-Let it be proposed to solve the system of equations

5x+4y=35,
7x-3y=6.

(1.)

(2.)

Multiplying equation (1) by 3, and equation (2) by 4, we have

Adding (3) and (4), member to member (Axiom 1), we have

whence

43x=129;
x=3.

(5.)

(6.)

We may now deduce the value of y by substituting the value of x in one of the original equations. Taking the first for example, we have

whence

15+4y=35;

4y=20,

149. In the same way, an unknown quantity may be eliminated from any two simultaneous equations. This method is expressed in the following

RULE.

Multiply or divide the equations, if necessary, in such a manner that one of the unknown quantities shall have the same coefficient in

both. Then subtract one equation from the other if the signs of these coefficients are alike, or add them together if the signs are • unlike.

In solving the preceding equations, we multiplied both members of each by the coefficient of the quantity to be eliminated in the other equation; but if the coefficients of the letter to be eliminated have any common factor, we may accomplish the same object by the use of smaller multipliers. In such cases, find the least common multiple of the coefficients of the letter to be eliminated, and divide this multiple by each coefficient; the quotients will be the least multipliers which we can employ.

150. Elimination by Substitution.--Take the same equations as before:

5x+4y=35,

7x-3y=6.

(1.)

(2.)

Finding from (1) the value of y in terms of x, we have

Substituting this value of y in (2), we have

(3.)

Substituting this value of x in (3), we have

y=5.

The method thus exemplified is expressed in the following

RULE.

Find an expression for the value of one of the unknown quantities in one of the equations; then substitute this value for that quantity in the other equation.

151. Elimination by Comparison.-Take the same equations

as before:

5x+4y=35,
7x-3y=6.

(1.)
(2.)

Derive from each equation an expression for y in terms of x,

Placing these two values equal to each other, we have

Find an expression for the value of the same unknown quantity in each of the equations, and form a new equation by placing these values equal to each other.

In the solution of simultaneous equations, either of the preceding methods can be used, as may be most convenient, and each method has its advantages in particular cases. Generally, however, the last two methods give rise to fractional expressions, which occasion inconvenience in practice, while the first method is not liable to this objection. When the coefficient of one of the unknown quantities in one of the equations is equal to unity, this inconvenience does not occur, and the method by substitution may be preferable; the first will, however, commonly be found most convenient.

1. Given {11x+3y=100

4x-7y=4

EXAMPLES.

to find the values of x and y. Ans. x=8; y=4.