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109. INDEPENDENT EQUATIONS are such as cannot be derived from one another, or reduced to the same form.

X y
2

Thus, x+y=10, +2=5, and 4x + 3y = 40 — y are not independent equations, since any one of the three can be derived from any other one; or they can all be reduced to the form x+y= But x+y= 10 and

10.

4xy are independent equations.

110. To find the value of several unknown quantities, there must be as many independent equations in which the unknown quantities occur as there are quantities.

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From the equation x+y

= 10

=

unknown

= 10 we cannot determine the value

of either x or y in known terms. If y is transposed, we have
y; but since y is unknown, we have not determined the
value of x. We may suppose y equal to any number whatever,
and then x would equal the remainder obtained by subtracting y
from 10. It is only required by the equation that the sum of two
numbers shall equal 10; but there is an infinite number of pairs
of numbers whose sum is equal to 10. But if we have also the
equation 4x y, we may put this value of y in the first equation,
x+y=10, and obtain x + 4x 10, or x = 2; then 4x=
8=y,
and we have the value of each of the unknown quantities.

=

ELIMINATION.

111. ELIMINATION is the method of deriving from the given equations a new equation, or equations, containing one (or more) less unknown quantity. The unknown quantity thus excluded is said to be eliminated.

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Transposing 4 x in (1) and dividing by 5, we have (3), which gives an expression for the value of y. Substituting this value of y in (2), we have (4), which contains but one unknown quantity; i. e. y has been eliminated. Reducing (4) we obtain (6), or Substituting this value of x in (3), we obtain (7), or Hence,

x = 2.

y

= 3.

RULE.

Find an expression for the value of one of the unknown quantities in one of the equations, and substitute this value for the same unknown quantity in the other equation.

NOTE. After eliminating, the resulting equation is reduced by the rule in Art. 102. The value of the unknown quantity thus found must be substituted in one of the equations containing the two unknown quantities, and this reduced by the rule in Art. 102.

Find the values of x and y in the following equations: —

2. Given (x+y=17) 172. 1x — y = 35

Ans.

x = 10.

y=

7.

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Finding an expression for the value of x from both (1) and (2), we have (3) and (4). Placing these two values of x equal to each other (Art. 13, Ax. 8), we form (5), which contains but one unknown quantity. Reducing (5) we obtain (7), or y = 5. Substituting this value of y in (3), we have (8), or x = 16. Hence,

RULE.

Find an expression for the value of the same unknown quantity from each equation, and put these expressions equal

to each other.

By this method of elimination find the values of x and y in the following equations:

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If we multiply (1) by 2, and (2) by 3, we have (3) and (4), in which the coefficients of x are equal; subtracting (4) from (3), we have (5), which contains but one unknown quantity. Reducing (5), we have (6), or y 1; substituting this value of y in (2), we obtain (7), which reduced gives (8), or x = 3.

=

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If we multiply (1) by 2, we have (3), an equation in which y has the same coefficient as in (2); since the signs of y are different in (2) and (3), if we add these two equations together, we have (4), which contains but one unknown quantity. Reducing (4), we have (5), or x = 18. Substituting this value of x in (1), we have (6), which reduced gives (7), or y -12. Hence,

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