5. Given u + 2x + 1 = 16, to find the value of x. Here x +1 = 4, by extracting the square root of each side . And therefore, by transposition, = 4–1 = 3. 6. Given 5ax 36 2dx + c, to find the value of x.

Here 5ax 2dx =c+ 3b; or (5a - 2d)x=ct3b; and therefore, by division, x=

5a 2d*

Here x

3+

17. Given + 10, to find the value of x.

2 3 4
2x 2x

6x Here a

+
= 20; and 3x

2x +

60; or 12x 3 4 8x + 6x 240; whence 10x = 240, or x = 24. 3

19 8. Given + 20

to find the valueof x. 2 3

2 2x

40 - X +- 19; or 3x -- 9 +2x = 120

3 3x + 57; whence 3x + 2x + 3x = 120+-57 + 9; that is, 8x

186, or x = 231

2x 9. Given v 3+5= 7, to find the value of x. 2x

2x Here ✓

= 7-5=2; whence, by squaring, 22 3

3 = 4, and 2x 12, or x =

='6.

2a2 10. Given x + V (a + x^)

to find the value V (a2 + x^)

of x.

wo!

a

Here x V (69 + x) + a2 + c = 2a”; or mv (az + xoa) ca —**, and 22. (am + xo) = a* 2a+32 + 204; whence aạxa + 204 = a4 20ʻx2 + x4, and aʻxa a4 2a*xa; therefore at a

aa 3a*x* = a*, or 22

and consequently a = V 3a2

3 3 -=av

3 V 3, the answer required. 9

EXAMPLES FOR PRACTICE. 1. Given 3x -- 2 + 24 = 31, to find the value of x.

Ans. x=3. 2. Given 4

9y = 14
14 - Ily, to find the value of

y.

Ans. y = 5. 3 Given x + 18 = 3x - 5, to find the value of x.

Ans. 2 =

115. 4. Given a tot 11, to determine the value of x. 2 3

Ans. x = 6. 5. Given 2x

+1= 5x – 2, to find the value of x. 2

6 Ans. X=

7 7

to determine the value of x.. 3 4 10

1 Ans. X=

1

4

* *3

X - 5 17. Given +

4

to find the value of x. 2 3 4

6 Ans. x = 3

13 8. Given 2 + V 3x v (4 + 5x), to find the value of x.

Ans. X =

12. 202 9. Given x ta=

to find the value of x. at x

Ans. x =

1 ola

2α

a

of a

a

+

pleanno

10. Given Vx+v(a + x) =

to find the value v (a + x)'

Ans. x =

3* ax - b

a

ba bx 11. Given 十

to find the value of x. 4 3 2 3

33 Ans. =

За. 26 12. Given v (as + x) = V (54 + x4), to find the value of a.

64 at Ans. x =V

2aa 13. Given v (a + x) + V (a - 2) = V ax, to find the value

402 Ans. =

a®+4

of a.

of x.

Ans. 3 =

15. Given a te=V[a3 + xV(62 + **)], to find the value

72

40 16. Given IV (2? + 3a%) - Av(x - 30%) = x V a, to find the value of x.

Ans. x = v

17. Given v (a + x) +- vla - x)=b, to find the value of it

b Ans. x =

V(4a - 6).

2 18. Given V (a + x) + V (a- x)=b, to find the value of a

23 Ans. x =

36 19. Given va tv x= vax, to find the value of x.

a

a

20. Given v

tov

Ag to determine the

2 + value of x.

Ans. 2 =

v cca

(-4) 21. Given v(a+-ax)=a-v (a? — ax), to find the value of x'.

Ans. x=3 v 3. 22. Given v(a -- 2) + xy (a’ – 1) = av (1 --"), to find the value of .

a2 Ans. x =

V

ac + 3 23 Given v(x + a)=(-V (a + b), to find the value of x.

10 +6 Ans. x =

-3. 2c

Of the resolution of simple equations, containing two unknown

quantities. When there are two unkuown quantities, and two independent simple equations involving them, they may be reduced to one, by any of the three following rules :

RULE 1.–Observe which of the unknown quantities is the least involved, and find its value in each of the equations, by the methods already explained; then let the two values, thus found, be put equal to each other, and there will arise a new equation with only one unknown quantity in it, the value of which may be found as before.

*

* This rule depends upon the well-known axiom, that things which are equal to the same thing, are equal to each other; and the two following methods are founded on principles which are equally simple and obvious.

EXAMPLES.

}

a

| 2x + 3y = 23 1. Given

to find the values of a and

y. ( 5x – 2) = 10

23 Зу Here, from the first equation, x =

2

10 + 2y And from the second, x =

;

5 23 Зу

10 + 2y Whence we have

2

5 Or 115 — 15y = 20 + 4y, or 19y

-- 115 - 20 = 95. 95

23 - 15 That is, y 5, and x=

4. 19

2 x + y 2. Given

to find the values of a and y
Y

b
Here, from the first equation, a = a- Y,

And from the second x = b to y,
Whence a “y=b+y, or 2y

= a - b,

b And therefore y

and x = a y. 2

- 5 Or by substitution, x = a

2

2 S &+y=72

to find the values of x and 十言

2y Here, from the first equation, x = 14

3

Зу
And from the second, x = 24

2
2y

Зу
Therefore, by equality, 14 - 24-

3

2

a

a

atb

3. Given 1x + y=85

9y

And consequently, 42 – 2y = 72

2'
Or, by multiplication, 84 - 4y= 144 - 9y;
And, therefore, also 5y = 144 – 84 = 60,
60

24
Or, by division, y 12, and x = 14
5

3

6.

At

EXAMPLES FOR PRACTICE.

1. Given 4x +y= 34, and 4y + x = 16, to find the values of x and Y.

Ans. x= 8, y

8, y = 2. 2. Given 2x + 3y = 16, and 3x – 2y = 11, to find the values of x and y.

Ans. x = 5, y = 2.

2x Зу
9 3х

2y

61 3. Given + and

to find the 5 4

20

4 5 120 values of x and y.

Ans. x = 1, y=s. 국 + 2y to find x and

y.

3 Ans. x = a + b, and y=1a-3

4. Given *** +27

to find x and y.

Ans. x = 12, and y=6.

+

80,

2

9 6. Given 2 3

to find x and y. X: 7::4:3

Ans. 2 =

12, and y :9.

20 Зу 17. Given x+y=

and
to find a and

y.
3 4

Ans. * = 42 17, and y 37 14. 8 Given y - 6 =

and x =

y + 6, to find a and y.

Ans. x = 24, and y=18. RULE 2.-Find the values of either of the unknown quantities in that equation in which it is the least involved ; then substitute this value in the place of its equal in the other equation, and there will arise a new equation with only one unknown quantity in it; the value of which may be found as before.

EXAMPLES.

x + 2y 17 1.

to find the values of ac and 23

y. Y From the first equation, a = 17 – 2y; which value, being substituted for x, in the second,

gives 3 (17 - 2y).- Y=2, Or 51

6y - Y=2, or 7y= 51 2 = 49. Whence y

ny, and x = 17 -- 2y = 3.
7
x +y

13? 2. Given

to find the values of x and y.

3 From the first equation, x= 13-y; which value, being substituted for a, in the second, Gives 13

Y - =3, or 2y 13 - 3= 10.

10 Whence y

5, and x = 13 - y = 8.
2
2 : 7::0;b

to find the values of x and y. 十 y2 =ora

1. Given 3.0

49

{

C

3. Given {