Method 5

Solution of Some General Form Equations

This method os based on the Sutra "Paravartya Yojayet " which means 'transpose and apply ' . In this rule + is converted into - and vice versa and
* is converted into / and vice versa .

These formula although very simple and in the eye , even then not used by many so try this to speed up your calculations .

Type 1 :

Equations of the form ax +b = cx + d . (very simple one )

Direct answer can be obtained by the formula

x = d - b
a - c

Example 1 : - Find the value of x in the equation
3x + 5 = 2x + 7

Answer :
x = 7 - 5
3 - 2
=> x = 2.
Exercises : -
Find the value of x in the following equqtions : -
1.) 3x + 2 = 7x + 5
2.) 6x + 2 = 2x + 5
3.) 9x + 2 = 2x + 4
4.) 23x + 12 = 12x + 4
5.) 7x - 21 = 9x - 31


.

Type 2 :

Equations of the form (x +a )(x + b) = (x + c)(x + d)

The above is solved by the formula

x = c.d - a.b
(a + b) - (c + d)
Derivation is simple . Try it yourself .
Use it directly to solve .

Example 1 : - Find the value of x in the Equation
(x – 3) (x – 2 ) = (x + 1 ) (x + 2 ).

Answer : -

x = 1.2 - (-3).(-2)
(-3 + (-2)) - (1 + 2)

=> x = 2 - 6 = -4
-5 - 3 -8

=> x = 1 / 2

Exercises : -

Find the value of x in the following equqtions : -

1.) (x - 4)(x-2) = (x + 3)(x + 1)
2.) (x + 1)(x - 4) = (x + 9)(x - 1)
3.) (a - 7)(a + 1) = (a + 3)(a + 8)
4.) (y + 5)(y - 5) = (y +1)(y + 2)
5.) (x - 6)(x - 1) = ( x + 5)(x - 9)

Type 3 :

Equations of type m + n = 0
(x + a ) (x + b)

Take L.C.M and proceed.
m(x+b) + n (x+a) = 0
(x + a) (x +b)
mx + mb + nx + na = 0
(x + a)(x + b)
(m + n)x + mb + na = 0
Therefore

x = - (m.b + n.a)
(m + n)

Example : - Find the value of x in the following equation .
5 + 2 = 0
(x + 3) (x - 4)

Answer : -

Use the formula directly : -

x = - (5.(-4) + 2.3)
(5 + 2)
=> x = - (-20 + 6)
7
x = 14 / 7
x = 2

Exercises : -
Find the value of x in the following equqtions : -
1.) 4 + 9 = 0
(x + 3) (x + 2)
2.) 6 + 5 = 0
(x - 5) (x + 2)
3.) 7 + 1 = 0
(x+1) (x + 9)

Type 4 :

Equations of the form (ax + b) = m
(cx + d) n

By cross – multiplication,
n ( ax + b) = m (cx + d)
nax + nb = mcx + md
nax - mcx = md – nb
x( na – mc ) = md – nb
x = (md - nb)
(na - mc)
Example : - Find the solution of x in equation
7.x + 2 = 5
3.x - 5 8

Answer : -

x = (5.(-5) - 8.2)
(8.7 - 5.3)
x = (-25 -16 )
(56 - 15 )
x = - 41
41
x = -1
Exercises : -

Find the value of x in the following equqtions : -

1.) 3x + 6 = 8
4x - 2 3
2.) 5x + 3 = 6
6x +8 4
3.) 9x + 7 = 2
7x - 3 5