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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

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

(3.)

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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, and we have

and

35–5

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(3.)

(4.)

Placing these two values equal to each other, we have

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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.

EXAMPLES.

1. Given {11x+3)=100}

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

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11. Given (12093=60x+152+} to find x and y.

Ans. x=7; y=93.

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Equations of the First Degree containing more than Two Unknown Quantities.

152. If we have three simultaneous equations containing three unknown quantities, we may, by the preceding methods, reduce two of the equations to one containing only two of the unknown quantities; then reduce the third equation and either of the former two to one containing the same two unknown quantities; and from the two equations thus obtained, the unknown quantities which they involve may be found. The third quantity may then be found by substituting these values in either of the proposed equations.

Take the system of equations

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

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Multiplying (1) by 5, and (3) by 2, we have

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Subtracting (8) from (7), 27y+14z=68.

(9.)

Multiplying (6) by 27, and (9) by 5, we have

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Substituting the values of y and z in (1),

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