4. Given 4x+3=34, and 4+16; to find x and y.

6. Given xs, and x2-y2=d; to find x and

Ans.

2

2

25

RULE 2.

1. Consider which of the unknown quantities you would first exterminate, and let its value be found in that equation, where it is least involved.

2. Substitute the value thus found for its equal in the other equation, and there will arise a new equation with only one unknown quantity, whose value may be found as before.

1. Given

EXAMPLES.

{ x+2y=17

3xy 2

; to find x and y.

From the first equation x=17-23,

And this value, substituted for x in the second, gives

17—23 X3—3—2,

Or 516-2, or 51732 ;

That is, 751—2—49;

Whence y=7, and x=17-23=17—14=3

12. Given

{} ; to find x and y.

From the first equation x=13—y,

And this value, being substituted for x in the second, Gives 13-y=3, or 13-2y=3;

That is, 2y13-3=10,

Or y5, and x=13-135-8.

3. Given

And this value of x, substituted in the second,

6. Given a b:: xy, and x3-y-d; to find x and y

Let the given equations be multiplied or divided by such numbers or quantities, as will make the term, which contains one of the unknown quantities, to be the same in both equations; and then by adding or subtracting the equations, according as is required, there will arise a new equation with only one unknown quantity, as before.

EXAMPLES.

1. Given

EXAMPLES.

S3x+5y=40
{ x+2y=14)

; to find x and y.

First, multiply the second equation by 3,
And we shall have 3x+6y=42;

Then, from this last equation subtract the first,
And it will give 6y-53-42-40, or y=2,
And therefore, x=14-2y=14-4=10.

2. Given

S5x-3y= 9
22x+5y=165

}

; to find x and y.

Let the first equation be multiplied by 2, and the second by 5,

And we shall have 10x 6y=18

10x+25y=80;

And if the former of these be subtracted from the latter, It will give 31y=62, or y== 2,

And consequently, x=2+3, by the first equation,

5

Multiply the first equation by 5, and the second by 3,

And we shall have 25x-15y=45

6x+15y=48;

Now, let these equations be added together,

And it will give 31x=93, or x=3,

4. Given ax+by=c, and dx+ey=f; to find x and y:

= 8, and

2

To exterminate three unknown quantities, or to reduce the three simple equations containing them to one.

RULE.

i. Let x, y and z, be the three unknown quantities to be exterminated.

2. Find the value of x from each of the three given equations.

3. Compare the first value of x with the second, and an equation will arise involving only y and z.

4. In like manner, compare the first value of x with the third, and another equation will arise involving only y and z.

5. Find the values of y and z from these two equations, according to the former rules, and x, y and z will be exterminated as required.

NOTE. Any number of unknown quantities may be exterminated in nearly the same manner, but there are offen

much