How to Determine the Upward Acceleration of M in the Pulley System?

[MARK 42]

In the arrangement shown in the figure, find the acceleration of M in upward direction. The pulleys are light. The strings are mass less and friction between the string and the pulleys is negligible.

The options are given. The final answer is (B) and this was the solution:

I don't understand the 1+7x2 step right there.
Also I fail to understand why there is 49m in the denominator of the solution instead of 7m (I'm assuming the equation is multiplied by 7. I might be wrong though.)
Can anyone explain me the solutions please? Thanks.

P.S.: Sorry about the low quality images, it was in a kind of jiffy.

 

[mfb]

The solution makes more sense without the line "1+7*2". The line above it gets multiplied by 7 and then plugged into the equation above it.

 

[MARK 42]

I don't understand the purpose of '1+7x2' either...

 

[MARK 42]

The solution makes more sense without the line "1+7*2". The line above it gets multiplied by 7 and then plugged into the equation above it.

Even if equation (ii) multiplied by 7, the equation must be something like this: 7T - 7mg = 7ma₂
So, on adding the two equations, 7T and -7T will get canceled out and the result will be Mg-7mg = Ma₁ + 7ma₁ ...(the acceleration is equal. therefore a₁ = a₂ .)
I don't understand where the 49ma₁ comes from.

 

[mfb]

The accelerations are not equal.

 

[MARK 42]

The accelerations are not equal.

Then how come they changed ma₂ to ma₁ in the fourth last step?
That way, the step should be written as Mg-7mg = Ma₁ + 7ma₂ right?

 

[mfb]

Then how come they changed ma₂ to ma₁ in the fourth last step?

They didn't, those are two different equations.

That way, the step should be written as Mg-7mg = Ma₁ + 7ma₂ right?

That is a possible way to write it.

Note the first two lines in the solution.

 

[MARK 42]

That is a possible way to write it.

Note the first two lines in the solution.

Oh, got it. Sorry, silly mistakes. Thanks you were really helpful.