Why does decrease in fixed taxes accompanied by a decrease in Government Spending of the same amount will decrease GDP?
The question basically boils down to whether the inequality below is true or not.
$$\bigg|\Delta G\bigg(\frac{1}{1-MPC}\bigg)\bigg|&gt;\bigg|\Delta T\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$
If $\Delta G=\Delta T$, we can divide by either and get
$$\bigg|\bigg(\frac{1}{1-MPC}\bigg)\bigg|&gt;\bigg|\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$
If we multiply each side by $1-MPC$ and we evaluate the absolute value, we get
$$1&gt;MPC$$ 
This statement is true, so that implies that the first inequality is true. By this inequality, we know that $\Delta GDP_G&gt;\Delta GDP_T$. Since $\Delta GDP_G&lt;0$ and $\Delta GDP_T&gt;0\implies \Delta GDP&lt;0$