Solving Equations

In this chapter we shall limit ourselves to solving linear equations in one variable only.

Any operation done on one side of an equation must be done on the other side also for the equation to remain true. This is the basic rule used to solve equations.

Transpose Method
We have also seen that the transpose method is the most convenient to solve equations. To apply this method the following rules are used:

1. A term in an equation that can be transferred or transposed from one side of the equation to the other by changing its sign.

Example :

If x - 12 = 5, then

x = 5 +  12;
If x + 1 = (- 2
x), then x + 2x = (- 1).

2. A number that divides a variable or a constant on one side, when transposed, multiplies on the other side, and vice versa.

Example :  = 10 becomes x = 10 x 5 = 50;
5x = 10 becomes x =  = 2

Using these rules, the variables are transposed to the left hand side, and the constant terms to the right hand side.

Let us see some examples of different kinds of linear equations.

Simple Equations

Example :

Solve 3a â€“ 5 = 16.

Solution :
 3 a â€“ 5 = 16 3 a = 16 + 5 Transpose - 5 to RHS 3 a = 21 Simplify RHS a = Transpose 3 to RHS a = 7 Simplify RHS

Equations with Variable on Both Sides
Example :

Solve x â€“ 3 = 5 + 5x

Solution :
 x â€“ 3 = 5 + 5 x x â€“ 5 x â€“ 3 = 5 Transpose 5 x to LHS -4 x -3 = 5 Simplify LHS -4 x = 5 + 3 Transpose (- 3) to RHS -4 x = 8 Simplify RHS X = Transpose (- 4) to RHS x = (-  2). Simplify RHS.

Check: LHS. = (- 2) -  3 =(- 5); RHS = 5 - 10= (- 5) = LHS

Complex Equations:

Example :

Solution:
 Simplify both RHS and LHS. Use cross multiplication 10(4a + 3) = 12(5 a â€“ 4) Use cross multiplication 40 a + 30 = 60 a - 48 Simplify both RHS and LHS 40 a â€“ 60 a = (-  48) -  30 Transpose 60 a to LHS and 30 to RHS (- 20) a = (- 78) Simplify RHS and LHS a = Transpose (- 20) to RHS a =    Check : LHS : == = =  RHS: =  = =  RHS = LHS Reduce the answer in to its lowest term.

Example :

= 2

Solution :
 1(5 x â€“ 1) = 2 (2 x +1) Apply cross multiplication 5 x â€“ 1 = 4 x + 2 Simplify RHS and LHS 5 x â€“ 4 x = 2+1 Transpose 4 x into LHS and (- 1) into RHS x = 3. Check: LHS. = Simplify RHS and LHS

Example :

Solution :
 Simplify LHS Simplify LHS 2(-  3x + 9) = 10(3 x -   9) Apply cross multiplication (-  6) x + 18 = 30 x -  90 Simplify LHS and RHS (-  6) x â€“ 30 x = -  90 -   18 Transpose 30 x to LHS and 18 to RHS (-  36) x = -108 Simplify RHS and LHS x = Transpose (- 36) into RHS x = 3 Check; LHS = -   = 1 - 1 = 0;            RHS = = 0 = LHS Simplify RHS

Example :

Solution :
 Given Simplify the numerator and denominator on LHS Simplify LHS by combining the like terms (- 2p) â€“ 4 = (-p) - 3 Apply cross multiplication (- 2p) + p = 4 - 3 Transpose (â€“ p) to LHS and (- 4) to RHS (- p) = 1 or p = (- 1) Check: LHS = RHS Simplify RHS and LHS

Example :

Solution :

(3y + 2) (8 y + 3) = (6 y + 1) (4 y + 1) Apply cross multiplication 24 y 2 + 9 y + 16 y + 6 = 24 y 2 + 6 y + 4 y + 1 Simplify RHS and LHS 24 y 2 + 9 y + 16 y - 24 y 2 - 6 y - 4 y = 1 - 6 Transpose 24 y 2, 6 y, 4y to LHS and 6 to RHS 15 y = (- 5) Simplify RHS and LHS y =  Transpose 15 to RHS y =  Simplify RHS

Check: LHS =  =  = = (- 3.)

RHS = = (- 3)

RHS = LHS