### First Method:

A term which occurs as a common factor in all the terms is equated to 0.

Examples:

12x+3x = 4x + 3x … As x is common factor both sides.So, x = 0.

9(x+3) = 4(x+3) … As (x+3) is common term both sides. So, x+3 = 0. x = -3.

### Second Method:

The product of the independent terms is same both sides then equated to 0.

Examples:

(x+5)(x+4) = (x+2)(x+10)

As product of independent terms (non-x terms): 5 x 4 = 2 x 10 , is same on both sides.

Therefore, x=0.

### Third Method:

The sum of the Denominators of two fractions having the same numerical numerator is equated to 0.

Examples:

1/(2x-1) + 1/(4x-1) = 0

Therefore, sum of denominators: 2x-1 + 4x-1 = 0

On solving x = 1/3

### Fourth Method:

The sum of the Numerators and the sum of the Denominators is the same, then that sum equated to 0.

Examples:

(2x +9)/ (2x +7) = (2x +7)/ (2x +9)

Here, Addition of both numerators = Addition of both Denominators.

Thus, 2x + 9 + 2x + 7 = 0

Hence 4x + 16 = 0 hence x = -4

If there is a numerical factor in the algebraic sum, then we remove that factor

(3x +4)/ (6x +7) = (x +1)/ (2x +3)

Here, Addition of both numerators = 4x +5

Addition of both Denominators = 8x + 10 =2(4x +5)

where, 2 is the numerical factor. So remove it

4x +5 =0. Hence x= -5/4

__NOTE:__

In above both examples, when we do cross multiplication the x2 term is getting cancelled. So instead of being quadratic eq and having 2 roots, it becomes simple linear equation and THUS has single root.

But if x^{2} term is not getting cancelled then it becomes quadratic equation and thus will have 2 roots/values. See next Meaning.

### Fifth Method:

Same Meaning as that of Type 4

Examples:

(3x +4)/ (6x +7) = (5x +6)/ (2x +3)

Here as well like previous meaning, Addition of both numerators = Addition of both Denominators. So x = -5/4

**BUT** on cross multiplication x^{2} term is not getting cancelled so it is quadratic equation and thus will have to find the 2nd root as well,

Calculated as |D1 – D2| = |N1 – N2| = 2x + 2 = 0(On removal of numerical factor)

So x = -1.

So Values of x are -5/4 and -1.

### Simultaneous equations-2

### ax + by = p

cx + dy = q

Solving,

x = (bq – pd) / (bc – ad)

y = (cp – aq) / (bc – ad)

Notice that for calculation of numerators (x any y) cyclic method is used and Denominators remains same for both x and y.

Examples:

2x + 3y =6

3x + 4y = 3

Applying above formula:

x = (9 – 24)/ (9 – 8) = -15

y = (18 – 6) (9 – 8) = 12

-3x + 5y = 2

4x + 3y = -5

Applying above formula:

x = (-25 -6) / (20+9) = -31/29

y = (8-15) / (20+9) = -7/29