The value of k for which the pair of equations 3x – y = 5 and 6x – 2y = k has infinitely many solutions is:

Step 1: Examining the System of Equations
Equation ₁: 3x – y = 5
Equation ₂: 6x – 2y = k

Step 2: Infinitely Many Solutions Condition
Infinitely many solutions are achieved when the equations define the same line.
That is, the equations should be scalar multiples of one another.

Step 3: Coefficient Comparison
– Equation ₁ coefficients:
– x coefficient: 3
– y coefficient: -1

– Equation ₂ coefficients:
– x coefficient: 6
– y coefficient: -2

Step 4: Checking Proportionality
Notice the proportionality of coefficients:
– x coefficient ratio: 6 ÷ 3 = 2
– y coefficient ratio: -2 ÷ (-1) = 2

Step 5: Constant Term Condition
For an infinite number of solutions, constant terms must also be under the same scaling.

Equations should be equal:
3x – y = 5
6x – 2y = k

Replacing the coefficient scaling:
– If the equations are for the same line, k should be 2 * 5 = 10

Step 6: Checking
If k = 10, the second equation is:
6x – 2y = 10
Divide by 2:
3x – y = 5 (Which is exactly the same as the first equation)

Conclusion:
The value of k for infinitely many solutions is 10.

Click here for more:
https://www.tiwariacademy.in/ncert-solutions/class-10/maths/