# Solve the following pair of linear equations by the elimination method and the substitution method

Question : Solve the following pair of linear equations by the elimination method and the substitution method :

(i)  x + y = 5  and   2x – 3y = 4

(ii)  3x + 4y = 10  and  2x – 2y = 2

(iii)  3x – 5y – 4 = 0  and  9x = 2y + 7

(iv)  $$x\over 2$$ + $$2y\over 3$$ = -1  and  x – $$y\over 3$$ = 3

## Solution :

(i)  By Elimination Method :

The given equations are

x + y = 5          ……(1)

and   2x – 3y = 4          …….(2)

Multiply equation (1) by 3 and adding with equation (2), we get

5x = 19   $$\implies$$  x = $$19\over 5$$

Now, Put the  value of x in equation (1), we get

$$19\over 5$$ + y = 5    $$\implies$$  y = $$6\over 5$$

Hence, x = $$19\over 5$$ and y = $$6\over 5$$

By Substitution Method :

The given equations are

x + y = 5          ……(1)

and   2x – 3y = 4          …….(2)

From equation (1),  y = 5 – x

Substituting the value of y in equation (2), we get

2x – 15 + 3x = 4     $$\implies$$   5x = 4 + 19

$$\implies$$  5x = 19       $$\implies$$    x = $$19\over 5$$

Now, Put the value of x in equation (1), we get

$$19\over 5$$ + y = 5    $$\implies$$  y = $$6\over 5$$

Hence, x = $$19\over 5$$ and y = $$6\over 5$$

(i)  By Elimination Method :

The given equations are

3x + 4y = 10          ……(1)

and   2x – 2y = 2         …….(2)

Multiply equation (2) by 2 and adding with equation (1), we get

7x = 14   $$\implies$$  x = 2

Now, Put the  value of x in equation (1), we get

3(2) + 4y = 10    $$\implies$$  4y = 10 – 6 = 4   $$\implies$$  y = 1

Hence, x = 2 and y = 1

By Substitution Method :

The given equations are

3x + 4y = 10          ……(1)

and   2x – 2y = 2          …….(2)

From equation (2),  y = x – 1

Substituting the value of y in equation (1), we get

3x + 4(x – 1) = 10

$$\implies$$  7x = 14       $$\implies$$    x = 2

Now, Put the value of x in equation (1), we get

3(2) + 4y = 10   $$\implies$$    y = 1

Hence,  x = 2 and y = 1

(iii)  By Elimination Method :

The given equations are

3x – 5y – 4 = 0    $$\implies$$   3x – 5y = 4          …..(1)

and  9x = 2y + 7  $$\implies$$  9x – 2y = 7         …….(2)

Multiplying equation (1) by 3 and subtracting equation (2) from equation (3), we get

-13y = 5        $$\implies$$        y = $$-5\over 13$$

Now, Put the value of y in equation (1), we get

3x – 5($$-5\over 13$$) = 4

$$\implies$$   3x = $$52 – 25\over 13$$

$$\implies$$  3x = $$27\over 13$$

$$\implies$$  x = $$9\over 13$$

Hence, x = $$9\over 13$$  and  y = $$-5\over 13$$

By Substitution Method :

The given equations are

3x – 5y – 4 = 0    $$\implies$$   3x – 5y = 4          …..(1)

and  9x = 2y + 7  $$\implies$$  9x – 2y = 7         …….(2)

From equation (2),    y = $$9x – 7\over 2$$

Substituting the value of y in equation (1), we get

3x – 5($$9x – 7\over 2$$) = 4     $$\implies$$    6x – 45x + 35 = 8

$$\implies$$  -39x = 8 – 35    $$\implies$$   -39x = -27

$$\implies$$  x = $$9\over 13$$

Now, Put the value of x in equation (2), we get

3 $$\times$$ $$9\over 13$$ – 5y = 4

$$\implies$$  5y = $$-25\over 13$$

$$\implies$$  y = $$-5\over 13$$

Hence, x = $$9\over 13$$  and  y = $$-5\over 13$$

(iv)  By Elimination Method :

The given equations are

$$x\over 2$$ + $$2y\over 3$$ = -1      $$\implies$$   3x + 4y = -6       ………(i)

and  x – $$y\over 3$$ = 3       $$\implies$$       3x – y = 9              ………..(ii)

Multiplying equation (2) by  4 and adding to equation (1), we get

15x = 30    $$\implies$$   x = 2

Now, put  x = 2 in equation (2), we get

3(2) – y  = 9    $$\implies$$   – y = 9 – 6 = 3

$$\implies$$  y = -3

Hence, x = 2 and y = -3

By Substitution Method :

The given equations are

$$x\over 2$$ + $$2y\over 3$$ = -1      $$\implies$$   3x + 4y = -6       ………(i)

and  x – $$y\over 3$$ = 3       $$\implies$$       3x – y = 9              ………..(ii)

From equation (2),  y = 3x – 9

Putting the value of y in equation (1), we get  y = 3x – 9

3x + 4(3x – 9) = -6     $$\implies$$    3x + 12x – 36 = -6

$$\implies$$   15x = 30   $$\implies$$  x = 2

Putting the value of x in equation (2), we get

3(2) – y = 9  $$\implies$$   – y = 9 – 6 = 3

$$\implies$$  y = -3

Hence, x = 2 and y = -3