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24. Given 2w+3x-2y+ z=15,)

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4x-y+3w-2z= 9, to find the values 3x+4y-2x- w=17, w, x, y, and z.

and 4y-5x+2x+4w=10,'

Ans. w=2, x=4, y=3, and z=5.

THE APPLICATION

Of Simple Equations of two unknown quantities, with their solutions.

1. Says A to B, if six years were added to my age, I should be as old as you; but if eight years were added to your age, you would be twice as old as me. Required each person's age.

Let x A's age in years,

and y= B's age in years;

Then x+6=y; and y+8=2x;

transposing the first Equation, y-x=6; transposing the second Equation, −y+2x=8;

.. by Addition,

hence, y=x+6=14+6=20;

.. A is 14, and B 20 years of age.

x=14.

2. Says A to B, if I had four of your marbles, I should have twice as many as you; but says B, if I had six of your marbles I should have eight times as many as you. Required each person's number.

Let x=the number A had,

and y=the number B had;

then x+4=the number A had after receiving four of B; and y—4—the number B had after giving four to A; ..x+4=2.(y-4), or x+4=2y-8;

by transposition, 2y-x=12;

but y+6=what B had after receiving six from A, and x-6=what A had after giving six to B; . y+6=8. (x—6), or y+6=8x-48;

... by transposition, 8x-y=54.

Multiplying this equation by 2,

16x-2y=108;

but 2y-x= 12;

by Addition, 15x=120; .. x=8; hence, 2y=x+12=8+12=20; .. y=10, . A had 8, and B 10.

3. A draper sold a certain quantity of cloth, consisting of blue and black, for £59. For the blue he charged 18s. and for the black 16s. per yard. He found, however, that if he had received 16s. for the blue, and 18s. for the black, per yard, he should have received £60. How many yards did he sell of each sort?

Let x=the number of yards of blue,
and y=the number of yards of black;
.. 18x=value of the blue,

and 16y value of the black;

but if, by the question, the blue was charged 16s. and the black 18s. per yard,

then 16x value of the blue,

and 18y-value of the black;
then 18x+16y=1180,
and 16x+18y=1200;

by Subtraction, -2x+2y= 20; multiplying this equation by 9, -18x+18y 180;

but 16x+18y=1200;

by Addition, 34x=1020; . x=30;

hence, 2y=20+2x=20+60=80;

y=40;

.. he sold 30 yards of blue, and 40 of black.

4. Bought a certain quantity of sugar, at 45s. per cwt., and a quantity of currants at 58s. per cwt., for £88. 10s., and sold the sugar at 2s. and the currants at 3s. per cwt. advance, and gained thereby £4. 5s. How many cwt. of each did he buy?

Let x=the number of cwts. of sugar,

and y=the number of cwts. of currants; then 45x=value of the sugar in shillings, and 58y value of the currants in shillings; .. 45x+58y=1770, (shillings in £88. 10s.) also, 2x the shillings gained by the sugar, and 3y=the shillings gained by the currants; ..2x+3y=85.

Multiplying this equation by 58, and the other by 3, 116x+174y=4930,

and 135x+174y=5310;

by Subtraction, 19x

= 380; .. x=20;

hence, 3y=85-2x=85-40-45; ..y=15;

whence, he bought 20 cwt. of sugar, and 15 cwt. of currants.

5. What fraction is that, to the numerator of which if one be added, its value becomes one-half; but if three be added to the denominator, its value will be one-third?

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hence, y=2x+2=10+2=12, the denominator;

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6. A certain sum of money put out to interest for ten months, at a certain rate per cent. amounted to £361. 13s. 4d.; and in eighteen months it amounted to £371. Required the sum and rate per cent.

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multiplying this equation by 120,

120x+xy=43400; (A)

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multiplying this equation by 200,

200x+3xy=74200;

but 360x+3xy=130200(=3 times equation A)

by Subtraction, 160x = 56000; ..x=350; substituting this value of x in the equation A,

by transposition,

42000+350y=43400;
350y= 1400;

y=4;

whence, the sum put out is £350, at four per cent.

per annum.

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