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Transp. 15y and 4y and 28, gives 57 = 194;
And dividing by 19, gives 3 = y.

17 - 3y
Then x =

4. 2

14+2y. , 2d . , x =

5

28+ 4y
This subst. for x in the 1st, gives + 3y = 17;

5
Mult. by 5, gives 28 + 4y + 15y = 85;
Transpos. 28, gives 19y = 57;
And dividing by 19, gives y = 3.

14+2y
Then x =

= 4, as before. 5

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17 - 2x 3dly, in the 1st trans. 2.x and div. by 3, gives y =

3

34-4*
This subst. for y in the 2d, gives 5.7 - -= 14;

3
Multiplying by 3 gives 15.7 34 + 4.x = 42;
Transposing 34, gives 19.2.
And dividing by 19, gives * = 4.

17 - 2x
Hence y =

3, as before. 3

76;

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2

5% -14 4thly, in the 2d tr. 2y and 14 and div. by 2, gives y=

15.2 - 42 This substituted in the 1st, gives 2.0 +

=17;

2
Multiplying by 2, gives 19x - 42 =34;
Transposing 42, gives 19x = 76;
And dividing by 19, gives x = 4.

5.X – 14
Hence y =

= 3, as before. 2

2. Given 2x + 3y = 29, and 3.x - 2y = 11; to find x

Ans. I = 7, and y = 5.

and g.

3. Given {+;}; to find x and y.

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Ans. x = 8, and y = 6.

4. Given

3:2

4. Giveni firmy

I re- gute = 20} ; to find x and y.

Ans. tô, and y = 4.

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+ 3y = 21, and 3 + 3r = 29; to find x

3

and y.

Ans. X = 9, and y = 6.

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§

+ 4, and

r +

4.

у 6. Given 10

2 = 3 3y

1; to find x and y. Ans. x = 8, and y = 6. 5 7. Given x : y:: 4:3, and 43 — 4,3 = 37; to find x and y.

Ans. ix = 4, and y = 3.

RULE III.

Let the given equations be so multiplied, or divided, &c, and by such numbers or quantities, as will make the terms which contain one of the unknown quantities the same in both equations; if they are not the same when first proposed.

Then by adding or subtracting the equations, according as the signs may require, there will remain a new equation, with only one unknown quantity, as before. That is, add the two equations when the signs are unlike, but subtract them when the signs are alike, to cancel that common term.

Note. To make two unequal terms become equal, as above, multiply each term by the co-efficient of the other.

EXAMPLES.

5x

= 9
Given 2.0 + 5y = 16 ) ; to find and

x y. Here we may either niake the two first terms, containing %, equal, or the two 2d terms, containing y, equal. To make the two first terms equal, we must multiply the 1st equation by 2, and the 2d by 5; but to make the two 2d terms equal, we must multiply the 1st equation by 5, and the 2d by 3; as follows.

1. By making the two first terms equal :

Mult. the 1st equ. by 2, gives 10.7 by = 18;
And mult. the 2d by 5, gives 10.6 +25y = 80;
Subtr. the upper from the under, gives 31y = 62;
And dividing by 31, gives

9 + 3y
Hence, from the 1st given equ. X =
2. By making the two 2d terms equal:

Mult. the 1st equat. by 5, gives 25.x - 15y = 45;
And mult. the 2d by 3, gives 6.X + 15y = 48;
Adding these two, gives

31.1. = 93;
And dividing by 31, gives

3.

5.7 9 Hence, from the 1st equ. y =

2.

5

MISCELLANEOUS EXAMPLES.

* +8 1. Given =

5.1 23 4

3 find .x and y.

Ans. x = 4, and y = 3. 3х

^x 2. Given + 10 = 13, and

= 4

2 to find x and yo

Ans. * = 5, and y = 3.

6.7 3. Given

+ = 10, and 5

3

6 to find x and y.

Ans. * = 8, and y = 4, 4. Given 3x + 4y = 38, and 4.x - 3y = 9; to find x and y.

Ans. x = 6, and 5.

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2y

+

b=14

14;

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PROBLEM II.

To Exterminate Three or More Unknown Quantities; Or, to

Reduce the Simple Equations, containing them, to a Single

one.

RULE.

This may be done by any of the three methods in the last problem: viz.

1. AFTER the manner of the first rule in the last problem, find the value of one of the unknown letters in each of the given equations : next put two of these values equal to each other, and then one of these and a third value equal, and so on for all the values of it ; which gives a new set of equations,

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with which the same process is to be repeated, and so on till there is only one equation, to be reduced by the rules for a single equation.

2. Or, as in the 2d rule of the same problem, find the value of one of the unknown quantities in one of the equations only; then substitute this value instead of it in the other equations; which gives a new set of equations to be resolved as before, by repeating the operation.

3. Or, as in the 3d rule, reduce the equations, by multiplying or dividing them, so as to make some of the terms to agree : then, by adding or subtracting them, as the signs may require, one of the letters may be exterminated, &c, as before.

EXAMPLES

9

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2y

3%,

16 - 2y

4%;

x + y + z= 1. Given * + 2y + 32

=16

; to find x, y, and e. * + 3y + 4z = 21 ) 1. By the 1st method : Transp. the terms containing y and zin each equa. gives

9

y ,
* 16
X = 21

3y - 42; Then putting the 1st and 2d values equal, and the 2d and 3d values equal, give 9 % = 16

2y 3z,

32 21 3y
In the Ist trans. 9, %, and 2y, gives y = 7 - 22;
In the 2d trans. 16, 3z, and 3y, gives y = 5 - Z;
Putting these two equal, gives 5-23 7-22;

Trans. 5 and 2z, gives z = 2.
Hence y = 5-%= 3, and x = 9-y-z = 4.
2dly. By the 2d method:

From the 1st equa. x = 9-y-z;
This value of x substit. in the 2d and 3d, gives

9 + y + 2z = 16,

9 + 2y + 32 = 21;
In the 1st trans. 9 and 2z, gives y
This substit. in the last, gives 23 z = 21;
Trans. % and 21, gives

2 = 2.
Hence again y = 7--23 = 3, and x = 9-y-z = 4.

3dly. By

=7

2z;

38

3dly. By the 3d method : subtracting the 1st equ. from the 2d, and the 2d from the 3d, gives

y + 2% = 7,

y + z = 5;
Subtr. the latter from the former, gives z = 2.
Hence y 5—% = : 3, and x = - 9-y-2 = 4.

2 + y + z = 18 2. Given x + 3y + 2z s to find x, y, and z. x + ty + z = 10 Ans. x = 4, y

6,= 8. x + y + z

= 27 3. Given

x + 3y + z = 20°; to find x, y, and z. x + 4y + šz

Ans. x = 1, y = 20, z = 60. 4. Given x - y = 2, x - x 3, and y - x =l; to find x, y, and z.

Ans. r = 7; y= 5; z = 4. $2,5

2.x + 3y + 4z = 34 5. Given

3.0 + 4y + 5z ; to find x, y, and z. 4x + 5y + 6z = 58

= 16

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= 46

A COLLECTION OF QUESTIONS PRODUCING SIMPLE

EQUATIONS.

X

Quest. 1. To find two numbers, such, that their sum shall be 10, and their difference 6. Let x denote the greater nunber, and y the less *.

Then, by the 1st condition x + y = 10,
And by the 2d

y = 6,
Transp. y in each, gives

X = 10

and x =
Put these two values equal, gives 6 +y = 10 -- y;
Transpos. 6 and - y, gives 2y = 4;
Dividing by 2, gives

у = 2.
And hence

X = 6 +y = 8.

9 Y,

6 + y;

* In all these solutions, as many unknown letters are always used as there are unknown numbers to be found, purposely the better to exercise the modes of reducing the equations : avoiding the short ways of notation, which, though giving a shorter solution, are for that reason less useful to the pupil, as affording less exercise in practising the several rules in reducing equations.

QUEST. 2

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