I am studying Essential Mathematics for Economic Analysis and in chapter 12 problem 10 there is a problem that I can solve but I do not understand why the solution works.
The problem asks us to differentiate the following system from consumer demand theory:
$$U'_1(x_1,x_2) = \lambda p_1 \\ U'_2(x_1,x_2) = \lambda p_2 \\ p_1 x_1 + p_2x_2=m $$
where system defines $x_1, x_2$ and $\lambda$ as differentiable functions functions of $p_1,p_2$ and $m$. We are asked afterwards find expression for $\partial x_1 / \partial p_1$.
Now I have no problem with the first step - the differentiated system will be:
$$U^{''}_{11}(x_1,x_2)dx_1 + U^{''}_{12}(x_1,x_2)dx_2 = \lambda dp_1 + p_1 d\lambda \\ U{''}_{21}(x_1,x_2)dx_1 + U{''}_{22}(x_1,x_2)dx_2 = \lambda dp_2 + p_2 d\lambda \\ p_1 dx_1 + x_1 dp_1 + p_2 d x_2 + x_2 dp_2=dm $$
Now my first attempt was to solve 2nd equation for $dx_2$ substitute it into the first and solve for $d x_1 /dp_1$ but I got wrong result. Afterwards in the solution in the back there is a hint that says that to solve this we should put $dp_2=dm=0$ after I follow this hint I get the correct solution:
$$\partial x_1 / \partial p_1 = \frac{[\lambda p_2^2 + x_1(p_2 U^{''}_{12}-p_1U^{''}_{21})]}{(p_1^2 U^{''}_{22} -2p_1p_2U^{''}_{12} + p_2^2 U^{''}_{11})}$$
But I don't get why we can simply assume $d p_2 = dm = 0$ that seems so arbitrary, how come that is necessary to find the correct solution?