Maths/physics mechanics question help

A light inextensible string passes over smooth pulley with mass 7kg and 3kg at each end.

By solving 2 simultonious equations, find acceleration of the system and the tention in the string
step by step pls

7 kg moves down
3 kg moves up
let, acceleration = a
and, tension = T

at 7 kg end,
7g - T = 7a .....(1)

at 3 kg end,
T - 3g = 3a ......(2)

4g = 10a
a = 0.4g = 0.4*9.8 = 3.92 m/s^2

T = 3a + 3g = 3*(9.8+3.92) = 3*13.72 = 41.16 N

ok it's simple. the formula for acceleration can be derived as:
m1=7kg (heavy)
m2=3kg(light)

heavy mass will move downward due to unbalanced force (unequal tension and weight)
weight will pull the body downward i.e. weight is greater than tension. so
F = W - T
m1a = m1g - T ------(1)

and for lighter mass. tension will be more to pull it upward
F = T - W
m2a = T - m2g ------(2)

now for acceleration simply add both equations so that tension will cancel out from equations.
result will be
a = (m1-m2) / (m1+m2)

for tension divide both equations to cancel out acceleration.
result:

T = (2*m1*m2*g) / (m1+m2)

acceleration will be 0.4m/s^2

tension will be 41.16 N

The tension T in the string is the same at both ends, so look at the forces on
each mass separately:

1.) for the 7 kg mass we know that since this is the heavier mass it will be
descending with acceleration a, and so the force equation will be 7*g - T = 7*a;

2.) the 3 kg mass will be ascending with acceleration a, so the force equation
will be 3*g - T = -3*a.

So we just subtract the second equation from the first to eliminate T:

7*g - T = 7*a
3*g - T = -3*a
-------------------
4*g = 10*a ----> a = (2/5)*g.

So T = 7*(g - a) = 7*(g - (2/5)*g) = 7*(3/5)*g = (21/5)*g.

Using F = ma we get:

7g - T = 7a...(1)

T - 4g = 4a...(2).....where a is the acceleration, T is the tension and g is acceleration due to gravity.

Adding gives:

3g = 11a

i.e. a = 3g/11 ms^-2

so, T = 4(a + g) = 4(14g/11) = 56g/11 Newtons

:)>

f = ma
f1 = 7a
f2 = -3a
net = 4a
a = 9.8 m^2 (what else could it be with unequal masses on a pulley?)
Tension = 9.8m/s^2(7 + 3kg) = 98N