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MATHMAA

# Substitution-Method

### Concept of Substitution Method

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.

#### Examples :-

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