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(Clearing of fractions) (b) 3x+96-2x=108.

(Transposing)

(Uniting)

Then

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351. In the solution of problems where we have two unknown quantities, we must

1. Form at least two independent equations from the statements of the problems.

2. So combine these equations that we may eliminate one of the unknown quantities.

3. After finding the value of one unknown quantity, we substitute this value for the quantity in one of the equations of the statement, and thus find the value of the other unknown quantity.

352. Elimination may be made by several methods, but only one need be outlined here, viz: Elimination by addition and subtraction. By this method

we

1. Multiply or divide the equations by such numbers as to cause each equation to contain the same number of one of the unknown quantities.

2. Then by adding or subtracting the equations thus formed we eliminate this unknown quantity.

Note. If the signs of the quantity to be eliminated are alike, we subtract the equations to eliminate the quantity.

If the signs of the quantity are one plus and the other minus, we add the equations to eliminate the quantity.

Examples of Combinations necessary for Elimination 1. (a) 3x-y=7

(b)

x-y=1 (Subtract to eliminate y.)

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2. (a) 3x-y= 7

(b) _x+y= 5 (Add to eliminate y.)

4x = 12

x= 3

Exercises with Equations of Two Unknown Quantities

353. Find the value of x and y:

1. (a) 2x+4y = 28.
(b) 4x+3y=31.

2. (a) 3x+y=41.
(b) 7x-y=79.
3. (a) 2x+2y= 26.
(b) 7x-y=67.
4. (a) 5x-2y= 8.
(b) 7x- y = 13.
5. (a) 9x+2y= 20.
(b) 3x + y = 7.

6. (a) 7x+4y = 40.
(b) 3x+2y= 18.
7. (a) 11x+8y = 37.
(b) 5x+6y= 18.
8. (a) 5x+2y=49.
(b) 3x-2y=23.
9. (a) y+2x=18.
(b) y-2x= 2.
10. (a) 7x+4y = 58.
(b) 9x-4y=38.

11. (a) 4x+3y = 22.
(b) -4x+2y= -12.

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354. In the following exercises, first clear the equations of fractions and then eliminate by addi

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Problems with Two Unknown Quantities

355. 1. A man paid $44 for 6 barrels of flour and 4 bushels of oats; and $32 for 3 barrels of flour and 7 bushels of oats.

of each ?

What were the prices

SUGGESTION. Let x= the price of the flour,

and

y= the price of the oats.

Then (a) 6x+4y=$ 44.

(b) 3x + 7y = $ 32.

2. What fraction is that, to the numerator of which, if 4 be added, the value is ; but if 7 be added to its denominator, its value is

Let

and

Then (a)

(b)

SUGGESTION

?

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3. What numbers are those whose difference is 7 and sum 33?

4. Divide 75 into two such parts that three times the greater will exceed seven times the less by 15.

5. If 7 bushels of wheat and 10 bushels of rye cost $15; and 4 bushels of wheat and 5 bushels of rye cost $8, what are the prices of each?

6. Half the sum of two numbers is 20, and three times their difference is 18. Find the numbers.

7. 1000 men and women are employed in a factory. The average daily wage of the men is $2.50 and of the women $1.50. If the wages amount to $2340 per day, how many men and how many women are employed?

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