FIND THE MISSING VALUE IF THE SYSTEM HAS NO SOLUTION

Find the value of k, so that the following system of equations has no solution:

Problem 1 :

3x + y = 1; (2k - 1)x + (k - 1)y = (2k - 1)

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

3/(2k - 1) = 1/(k-1) ≠ 1/(2k - 1)

3(k - 1) = 1(2k - 1)

3k - 3 = 2k - 1

3k - 2k = -1 + 3

 k = 2

Hence, the required value of k is 2.

Problem 2 :

3x + y = 1; (2k - 1)x + (k - 1)y = (2k + 1)

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

3/(2k - 1) = 1/(k-1) ≠ 1/(2k + 1)

3(k - 1) = 1(2k - 1)

3k - 3 = 2k - 1

3k - 2k = -1 + 3

 k = 2

Hence, the required value of k is 2.

Problem 3 :

x - 2y = 3; 3x + ky = 1

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

1/3 = -2/k ≠ -3/-1

1/3 = -2/k

k = -6

Hence, the required value of k is -6.

Problem 4 :

x + 2y = 5; 3x + ky + 15 = 0

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

1/3 = 2/k ≠ -5/15

1/3 = 2/k and 2/k ≠ -5/15

k = 6 and k ≠ -6

Hence, the required value of k is 6.

Problem 5 :

kx + 2y = 5; 3x - 4y = 10

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

k/3 = 2/-4 ≠ -5/-10

k/3 = 2/-4 and k/3 ≠ 1/2

k = -3/2 and k ≠ 3/2

Hence, the required value of k is -3/2.

Problem 6 :

x + 2y = 3; 5x + ky + 7 = 0

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

1/5 = 2/k ≠ -3/7

1/5 = 2/k and 2/k ≠ -3/7

k = 10 and k ≠ 14/-3

Hence, the required value of k is 10.

Problem 7 :

8x + 5y = 9; kx + 10y = 15

Solution :

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

8/k = 5/10 ≠ -9/-15

8/k = 1/2 ≠ 3/5

8/k = 1/2 and 8/k ≠ 3/5

k = 16 and k ≠ 40/3

Hence, the required value of k is 16.

Problem 8 :

Find the values of k for which the following system of equations is inconsistent

(3k + 1)x + 3y = 2

(k² + 1)x + (k - 2)y - 5 = 0

Solution :

(3k + 1)x + 3y = 2

(k² + 1)x + (k - 2)y - 5 = 0

Since the system has no solution, it is inconsistent.

Slopes will be equal but y-intercepts will not be equal.

m₁ = -(3k + 1) / 3 and m₂ = -(k² + 1)/(k - 2)

m₁ = m₂

(3k + 1)/3 = (k² + 1)/(k - 2)

(3k + 1)(k - 2) = 3(k² + 1)

3k² - 6k + k - 2 = 3k² + 3

-5k - 2 = 3

-5k = 3 + 2

-5k = 5

k = -1

So, for k = -1 the system is inconsistent 

Problem 9 :

-5x = 8 - ky

18y + 5x = -10x + 21

In the given system of equations, k is constant. If the system has no solution, what is the value of k ?

Solution :

-5x = 8 - ky ----(1)

18y + 5x = -10x + 21 -----(2)

From (1),

5x = ky - 8

ky = 5x + 8

y = (5/k) x + (8/k)

From (2),

18y = -10x - 5x + 21

18y = -15x + 21

y = (-15/18) x + (21/18)

y = (-5/6)x + 7/6

5/k = -5/6

k = -6

So, the value of k is -6.

Problem 10 :

(4/3) + (5x/4) = 28y + (5/8)x

my = (1/2) (5x - 8)

In the given system of equations, m is a constant. If the system has no solution, what is the value of m ?

Solution :

(4/3) + (5x/4) = 28y + (5/8)x -----(1)

my = (1/2) (5x - 8) -----(2)

28y = (5x/4) - (5x/8) + (4/3)

28y = (10x - 5x)/8 + (4/3)

28y = 5x/8 + (4/3)

y = -5x/224 - (4/84)

y = -5x/224 - (1/21)

Slope = -5/224

my = 5x/2 - 4

y = 5x/2m - (4/m)

Slope = -5/2m

-5/224 = -5/2m

2m = 224

m = 224/2

m = 112

So, the value of m is 112.

Problem 11 :

Which of the following systems of linear equations has no solution ?

Solution :

When two lines are parallel, they will have no solution.

Option a :

y = 8 and x = 8

The point of intersection of these two horizontal and vertical lines is (8, 8).

Option b :

y = 8, y = 8x + 8

Slope of the first line = 0

Slope of the second line = 8

Since slopes are not equal, they may be intersecting.

Option c :

y = 8x, y = 8x + 8

Slope of the first line = 8

Slope of the second line = 8

Since slopes are equal, they may be parallel.

So, option c is correct.