# (i) For which values of a and b will the following pair of linear equations have an infinite number of solutions?

2x + 3y = 7

(a - b) x + (a + b) y = 3a + b - 2

(ii) For which value of k will the following pair of linear equations have no solution?

3x + y = 1

(2k -1) x + (k -1) y = 2k +1

**Solution:**

(i)

2x + 3y - 7 = 0

(a - b) x + (a + b) y - (3a + b - 2) = 0

a_{1}/a_{2}= 2/(a - b)

b_{1}/b_{2}= 3/(a + b)

c_{1}/c_{2}= 7/(3a + b - 2)

For infinitely many solutions,

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

2/(a - b) = 7/(3a + b - 2)

6a + 2b - 4 = 7a - 7b

a - 9b = - 4 ....(1)

2/(a - b) = 3/(a + b)

2a + 2b = 3a - 3b

a - 5b = 0 ....(2)

Subtracting (1) from (2), we obtain

4b = 4

b = 1

Substituting b = 1 in equation (2), we obtain

a - 5 × 1 = 0

a = 5

Hence, a = 5 and b = 1 are the values for which the given equations give infinitely many solutions.

(ii)

3x + y - 1 = 0

(2k - 1) x + (k -1) y - 2k - 1 = 0

3x + y - 1 = 0

(2k -1) x + (k -1) y - 2k - 1 = 0

a_{1}/a_{2}= 3/2k - 1

b_{1}/b_{2}= 1/k - 1

c_{1}/c_{2}= -1/(-2k - 1) = 1/(2k + 1)

For no solution,

a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2}

3/2k - 1 = 1/k - 1 ≠ 1/(2k + 1)

3/2k - 1 = 1/k - 1

3k - 3 = 2k -1

k = 2

Hence, for k = 2 the given equation has no solution.

**Video Solution:**

## (i) For which values of a and b will the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a - b) x + (a + b) y = 3a + b - 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k -1) x + (k -1) y = 2k +1

### NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.5 Question 2:

(i) For which values of a and b will the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a - b) x + (a + b) y = 3a + b - 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k -1) x + (k -1) y = 2k +1

The values of a and b for which the equations 2x + 3y = 7 and (a - b) x + (a + b) y = 3a + b - 2 will have infinitely many solutions will be a = 5 and b = 1.

The value of k for wich the equations 3x + y = 1 and (2k -1) x + (k -1) y = 2k +1 will have no solution is k = 2.