Question

How many solution these pair of equations x-4y-6=0 and 3x-12y-18=0 will have?

Answer 1

Given :- How many solution these pair of equations x-4y-6=0 and 3x-12y-18=0 will have ?

concept used :-

• A linear equation in two variables represents a straight line in 2D Cartesian plane .

• If we consider two linear equations in two variables, say :-

➻ a1x + b1y + c1 = 0

➻ a2x + b2y + c2 = 0

Then :-

✪ Both the straight lines will coincide if :-

a1/a2 = b1/b2 = c1/c2

➻ In this case , the system will have infinitely many solutions.

➻ If a consistent system has an infinite number of solutions, it is dependent and consistent.

✪ Both the straight lines will be parallel if :-

a1/a2 = b1/b2 ≠ c1/c2.

➻ In this case , the system will have no solution.

➻ If a system has no solution, it is said to be inconsistent.

✪ Both the straight lines will intersect if :-

a1/a2 ≠ b1/b2.

➻ In this case , the system will have an unique solution.

➻ If a system has at least one solution, it is said to be consistent..

Solution :-

comparing the given equations x - 4y - 6 = 0 and 3x - 12y - 18 = 0 with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, we get,

• a1 = 1
• a2 = 3
• b1 = (-4)
• b2 = (-12)
• c1 = (-6)
• c2 = (-18)

Checking now,

→ a1/a2 = 1/3

→ b1/b2 = (-4)/(-12) = 1/3

→ c1/c2 = (-6)/(-18) = 1/3

therefore,

• a1/a2 = b1/b2 = c1/c2 = 1/3 .

Hence, we can conclude that, Both the straight lines will coincide if :-

• a1/a2 = b1/b2 = c1/c2
• In this case , the system will have infinitely many solutions.

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Answer 2

Step-by-step explanation:

The given lines seem to be parallel to each other since there gradient is same.

Let us first find the gradient of each line

Equation of linear line is

y = mx + c

Gradient of first line is

x - 4y -6 =0

x -6 = 4y

y = 1/4 x - 3/2

Here gradient is 1/4

Gradient of second line is

3x- 12y-18 = 0

3x -18 = 12y

y = 1/4x - 3/2

Here gradient is

1/4

Since the two gradients are same and y intercept are same so these lines will be at same levels and have infinite solutions