x + y = 14
x - y = 4

the equation have unique solution.
Pair of linear equations are consistent.

Here, a₁ = 3, b₁ = -1 and c₁ = -5
a₂ = 6, b₂ = -2 and c₂ = k


Hence the given system will have infinite no. of solution if k = -10.

2x - 3y = 4; 4x - 6y = 7; 6x - 9y - 12.
Sol. Pair of infinite solutions :
2x - 3y = 4; 6x - 9y = 12
Here,    a₁ = 2, b₁ = -3 and c₁ = 4
and    a₂ = 6, b₂ = -9 and c₂ = 12
The given values satisfy the condition :



So, given equations have infinite solutions pair of no solutions :
2x - 3y = 4; 4x - 6y = 7
Here,    a₁ = 2, b₁ = -3 and c₁ = 4
and    a₂ = 4, b₂ = -6 and c₂ = 7
The given values satisfy the condition:



So, given equations have no solutions.

We have,
kx - 4y = 3; 6x - 12y = 9
Here,    a₁ = k, b = -4,c₁ = 3
a² = 6, b₂ = - 12, c₂ = 9
For infinitely many solutions,

Hence, the given system of equation has infinitely many solution if, k = 2.

The given pair of equation is
2x + 3y = 5
5x - 2y = 3
Putting x = 1,y = 1 in eq. (i) and (ii), we get
2x + 3y = 5
⇒ 2(1) + 3(1) = 5
⇒ 5 = 5 which is true.
Also, 5a - 2y = 3
⇒ 5(1) - 2(1) = 3
⇒    3 = 3 which is true.
Thus, x = 1, y - 1 is the solution of the given pair of equations.