Explanation: The number of solutions of two linear equations representing intersecting lines is 1 because two linear equations representing intersecting lines has a unique solution.

Explanation:

Let the width be x.

Then length be x+4

According to the question

l+b=36

⇒ x+(x+4)=36

⇒ 2x+4=36

⇒ 2x=36-4

⇒ 2x=32

⇒ x=16.

Hence, The length of garden will be 20 m and width will be 16 m.

Area = length × breath = 20×16 = 320 m²

Explanation – The system of linear equations a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0 has no solution if the equation satisfy the condition a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Explanation

Let speed of boat = x km/h

speed of current = y km/h

∴ Downstream speed = (x+y) km/h

Upstream speed = (x−y) km/h

also,

Speed = Distance/Time

⇒ Time = Distance/Speed

According to question,

In downstream, 20/(x+y) = 2

x + y = 10 ….. (i)

In upstream, 4/(x-y) = 2

x – y = 2 ….. (ii)

Subtracting eq. (ii) from (i), we get

2y=8

⇒ y=4

Therefore, the speed of current is 4 km/h.

Explanation:

x + y = 8

⇒ x = 8-y …… (1)

⇒ x + y = 4.(x – y) ….. (2)

Substitute (1) in (2)

8= 4x-4y

⇒ 2=x-y

⇒ 2=8-y-y

⇒ 2y=8-2

⇒ y=3

therefore,

x=8-3=5

Hence, the required numbers are 5 and 3.

Explanation

Given:

a₁ =3, a₂ =6, b₁ =−4, b₂=−k, c₁=−7 and c₂=−5

If there is a unique solution, then

a₁/a₂ ≠ b₁/b₂

⇒ 3/6 ≠ -4/-k

⇒ -3k ≠ -4×6

⇒ k ≠ 8

Explanation:

Here, A=x, B=3x, C=y.

180= 4x+y …. (i) (Sum of angles of a triangles= x+3x+y)

180-4x=y ….. (ii)

Also, 3y-5x=30 …. (iii)

Substituting the value of (ii) in (iii)

3(180-4x)-5x=30

⇒ 540-12x-5x=30

⇒ -17x= 30-540

⇒ 17x= 510

⇒  x= 30 …. (iv)

But angle B = 3x

Therefore angle B = 3×30 = 90°

Explanation

a₁ = a, a₂ = p, b₁ = b, b₂ = q, c₁ = c and c₂ = r

Since the pair of given linear equations has a unique solution therefore,

∴ a₁/a₂ ≠ b₁/b₂

⇒ a/p ≠ b/q

⇒ aq ≠ bp