Use of Slutsky equation I know that the Slutsky equation is defined as:

$\frac{\partial x_1^s}{\partial p_1} = \frac{\partial x_1^m}{\partial p_1} + x_1^o \frac{\partial x_1^m}{\partial m}$

My problem is right now is making use of this information given (I am aware of how to take partial derivatives) but cannot seem to understand how to apply it to problem sets.

Here's an example (I'm more concerned about the steps on how to get to the answer not just the answer);

A consumer has preferences given by $U(x_1,x_2)= x_1^2x_2$

(a) Derive the demand curves for $x_1, x_2$ when prices and income are given by $p_1, p_2$ and $m$ 

$x_1^*=2m/3p_1$ and $x_2^*=m/3p_2$  -I think I understood how to do that

(b) Illustrate the equilibrium on a diagram when $p_1$ = $p_2$ $ =$ 1 and
$I$ = $12


the way I did this was by graphing and simply finding the equllibrium point graphically based on the Demands for goods $1$ and $2$ on the budget line


(c) Calculate the exact income and substitution effects for $x_1$ when $p_1$
rises to $3.

-The only way I'm currently able to do this is without Calculus, as described in this video which doesn't seem to sit well with me being that the Slutsky Equation is defined very clearly with use of calculus. I just don't know how to apply it.

(d) Explain your exact results using the appropriate Slutsky equation.


same problem here.


Note: I'm no simply looking for someone to "do my homework" I'm primary interest is in knowing how to apply the Slutsky equation when facing similar problems.
Utility function $u(x_1, x_2) = x_1^2x_2$.


  Q. Derive the demand for $x_1$ and $x_2$ as a function of $p_1$, $p_2$ and $m$.


Here are the demand functions for $x_1$ and $x_2$:
$$x_1(p_1, p_2, m) = \frac{2m}{3p_1}$$
$$x_2(p_1, p_2, m) = \frac{m}{3p_2}$$


  Q. Illustrate the equilibrium in a diagram when $p_1=1$, $p_2=1$ and $m=12$.





  Q. Suppose $p_1$ rises to 3. Calculate the substitution effect and income effect.


If $p_1$ rises to 3, the new equilibrium choice is $\left(\frac{8}{3}, 4\right)$. To find the Substitution effect and Income effect using Slutsky approach, we will find the equilibrium at new set of prices when the consumer has just enough money to buy the old equilibrium bundle i.e. we will find the demand at prices $(3,1)$ when income is $m' = 3(8) + 1(4) = 28$. Substituting this data in the demand functions, we get the equilibrium choice as: $\left(\frac{56}{9}, \frac{28}{3}\right)$. Here is how the situation looks in graph:


Substitution effect = $\displaystyle\frac{56}{9} - 8 = -\frac{16}{9}$

Income effect = $\displaystyle\frac{8}{3} - \frac{56}{9} = - \frac{32}{9}$


  Q. Explain your exact results using the appropriate Slutsky equation.


Slutsky equation:
Change in Demand =  Change in Demand due to substitution effect + Change in Demand due to income effect

$$\Delta x_1 = \Delta^s x_1 + \Delta^i x_1 = -\frac{16}{9} - \frac{32}{9} 
 = -\frac{16}{3}$$