A system does 200 J of work and at the same time absorbs 150 J of heat. The magnitude of the change in internal energy is _________J. (Nearest integer)

We recall the First Law of Thermodynamics, which is stated as $$\Delta U = Q - W,$$ where $$\Delta U$$ is the change in internal energy of the system, $$Q$$ is the heat absorbed by the system from the surroundings, and $$W$$ is the work done by the system on the surroundings.

According to the question, the system absorbs heat, so the heat term is positive. Numerically we have

$$Q = +150\ \text{J}.$$

The system also does work on the surroundings, therefore the work term counted in the equation is also positive. The data give

$$W = +200\ \text{J}.$$

Now we substitute these two values into the First Law expression:

$$\Delta U = Q - W = (+150\ \text{J}) - (+200\ \text{J}).$$

Carrying out the subtraction step by step, we first write the numerical difference,

$$\Delta U = 150\ \text{J} - 200\ \text{J}.$$

Performing the arithmetic,

$$\Delta U = -50\ \text{J}.$$

The question asks for the magnitude of the change in internal energy, that is, the absolute value:

$$\left|\Delta U\right| = \left|-50\ \text{J}\right| = 50\ \text{J}.$$

So, the answer is $$50\ \text{J}.$$