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(111.) The method thus exemplified is expressed in the following

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RULE. Find an expression for the value of the same unknown quantity in each of the equations, and form a new equation by putting one of these values equal to the other. Ex. 3. Find the values of x and y in the equations 2x+3y=22, and 5-5=3.

Ans. x=8, y=2. Ex. 4. Find the values of x and y in the equations 3x+y=14, and 5x-y=10.

Ans. x=3, y=5. Ex. 5. Find the values of x and y in the equations 5x+2y=43, and <= =5.

Ans. x=7, y=4. Ex. 6. Find the values of x and y in the equations 42+59=51, and six+y=13.

Ans. x=4, y=7. Ex. 7. Find the values of x and y in the equations 4–2y=12, and =2.

Ans. x=8, y=10. Ex. 8. Find the values of x and y in the equations 62—9y=33, and 5x-y=47.

Ans. x=10, y=3.

QUEST.-Give the rule for elimination by comparison.

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Ex. 9. Find the values of x and y in the equations 8x-3y=36, and +2=6.

Ans. x=9, y=12. Ex. 10. Find the values of x and y in the equations 72–4y=73, and +=7.

Ans. x=15, y=8. Ex. 11. A boy bought 3 apples and 5 oranges for 26 cents; he afterward bought, at the same rate, 4 apples and 7 oranges for 36 cents. How much were the apples and oranges apiece? Ans. The apples were 2 cents and the oranges 4

cents. . Ex. 12. A market-woman sells to one person 6 quinces and 4 melons for 72 cents; and to another, 4 quinces and 2 melons, at the same rate, for 40 cents. How much are the quinces and melons apiece? Ans. The quinces are 4 cents and the melons 12.

cents apiece. Ex. 13. In the market I find I can buy 4 bushels of barley and 3 bushels of oats for $2; and, at the same price, 8 bushels of barley and 1 bushel of oats for $3. What is the price of each per bushel ?

Ans. The barley is 35 cents and the oats 20 cents.

Ex. 14. Six yards of broadcloth and ten yards of taffeta cost $56; and, at the same rate, eight yards of broadcloth and 12 yards of taffeta cost $72. What is the price of a yard of each ?

Ans. The broadcloth cost $6 and the taffeta $2. Ex. 15. A person expends one dollar in apples and pears, buying his apples at 2 for a cent and his pears

at 2 cents a piece; afterward he accommodates his neighbor with of his apples and } of his pears for 38 cents. How many of each did he buy?

Ans. 56 apples and 36 pears.

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(112.) ELIMINATION BY ADDITION AND SUBTRACTION.

Ex. 1. To illustrate this method, take the two equa. tions

x+y=12,

3-y= 6. Since the coefficients of y in the two equations are equal, and have contrary signs, we may eliminate this letter by adding the two equations together, whence we obtain

2x=18,

x= 9. We may now deduce the value of y by substituting the value of x in one of the original equations. Substituting in the first equation, we have

9+y=12, whence

: y= 3. Since the coefficients of æ are equal in the two original equations, we might have eliminated this letter by subtracting one equation from the other. Sub. tracting the second from the first, we obtain

2y=6,

y=3. Ex. 2. Again : let us take the equations

2x+3y=13,

5x+4y=22. We perceive that if we could deduce from the pro. posed equations two other equations in which the co

efficients of y should be equal, the elimination of y might be etfocted by subtracting one of these new equations from the other.

It is easily seen that we shall obtain two equations of the form required if we multiply all the terms of each equation by the coefficient of y in the other. Multiplying, therefore, all the terms of equation first by 4, and all the terms of equation second by 3, they become

83 +12y=52,

15x4-12y=-66. Subtracting the former of these equations from the latter, we find

73= 14, whence

= 2. In like manner, in order to esiininate 2, multiply the first of the proposed equations by , and the second by 2; they will become

10x+15y=65,

10x+ 8y=44. Subtracting the latter of these two wquations fruir the former, we have

7y=21, whence

y= 3. (113.) This last method is expressed in the follow

ing

RULE. 1. Multiply or divide the equations, if necessary, in such a manner that one of the unknown quantities shall have the same coefficient in both.

QUEST.-Give the rule for elimination by addition and sabtraction.

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2. If the signs of the coefficients are the same in both equations, subtract one equation from the other; but, if the signs are unlike, add them together. Ex. 3. Find the values of x and y in the equations 3x+4y=59, and 5x+7y=102.

Ans. x=5, and y=11. Ex. 4. Find the values of x and y in the equations *=10, and +=11.

Ans. x=12, and y=30. Ex. 5. Find the values of x and y in the equations +y=16, and 2+ =12

Ans. x=10, y=14. Ex. 6. Find the values of x and y in the equations 5+2=7, and 3x—=44.

Ans. x=16, y=20. Ex. 7. Find the values of x and y in the equations

%=5, and +3=7.

Ans. x=14, y=10. Ex. 8. Find the values of x and y in the equations +>=21, and 3+5y=52.

Ans. x=16, y=10. Ex. 9. Find the values of u and y in the equations

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Ans. x=12, y=20. Ex. 10. Find the values of x and y in the equations

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