**MATHMAA**

*Substitution method* is one of the method to *solve* algebraic equations. To learn this *method* follow the following steps.

- First, we need to isolate any one of the variable i.e. ('x' or 'y') in the given two equations.
- Substitute this isolate variable value in other equation and
*solve*the variable . - Substitute
*solved*variable in any one of the equation to*solve*other variable.

Thus we can *solve* two equations.

To understand the* substitution method* better go through the following *examples* and *solutions*.

1. *Solve* the following equations.

2x - y - 3 = 0

4x - y - 5 = 0

Solution:-

The given equations are,

2x - y - 3 = 0 ----------(1)

4x - y - 5 = 0 ----------(2)

From (1) by isolating 'y' value, We get,

y = 2x - 3 ----------(3)

Plug (3) in (2), We get,

4x - (2x - 3) - 5 = 0

4x - 2x + 3 - 5 = 0

2x - 2 = 0

2x = 2

x = 1

Plug 'x' value in (3), We get,

y = 2(1) - 3

y = 2 - 3

y = -1

Therefore x = 1 and y = -1

2. Solve 2x + 3y - 9 = 0, 4x + 6y - 18 = 0 by *substitution method*.

Solution:-

The given equations are,

2x + 3y - 9 = 0 -----------(1)

4x + 6y - 18 = 0 -----------(2)

From (1), We get,

3y = 9 - 2x

y = \(\frac{9-2x}{3}\) ---------(3)

Substituting (3) in (2), We get,

4x + 6(\(\frac{9-2x}{3}\)) - 18 = 0

4x + 2(\(\frac{9-2x}{3}\)) - 18 = 0

4x + 18 - 4x - 18 = 0

0 = 0 ( This is a True statement)

Hence the equations contain infinitely many *solutions*. So to find the *solutions* we put x = k where 'k' is any real number.

Plug x = k in (3), We get,

y = \(\frac{9-2k}{3}\).

Therefore x=k and y = \(\frac{9-2k}{3}\).

3. Four years ago a mother was four times as old as her daughter. Six years later, the mother will be two and a half times as old as her daughter. Form the pair of linear equation for the situation and determine the present ages of mother and daughter in years, solving linear equations by *substitution method*.

Solution:-

Let the present age of mother = x years

Let the present age of daughter = y years

The age of mother

*SHARE YOU ENORMOUS EFFORT AND SMART EXAMPLES HERE*