How to calculate the intercept of an indifference curve in an Edgeworth box? Consider an exchange economy with two goods (Good 1 and 2) and two individuals ($A$ and $B$):


$A$'s utility function is $u_A(x_{A_1}, x_{A_2}) = 2x_{A_1} + 5x_{A_2}$ 
$B$'s utility function is $u_B(x_{B_1}, x_{B_2}) = 3x_{B_1} + 7x_{B_2}$ 
$A$'s initial endowment is $(\omega_{A_1}, \omega_{A_2}) = (4, 6)$
$B$'s initial endowment is $(\omega_{B_1}, \omega_{B_2}) = (10, 2)$


The core condition requires:

$$
2x_{A_1} + 5x_{A_2} \geq 2\omega_{A_1} + 5\omega_{A_2} = 2 \times 4 + 5 \times 6 = 38
$$

and

$$
3x_{B_1} + 7x_{B_2} \geq 3\omega_{B_1} + 7\omega_{B_2} = 3 \times 10 + 7 \times 2 = 44
$$

The line with  equation $2x_{A_1} + 5x_{A_2} = 38$ hits the left boundary at:

$$
(x_{A_1}, x_{A_2}) = (0,38/5), ~~~~(x_{B_1}, x_{B_2}) = (14, 2/5)
$$

The line with equation $3x_{B_1} + 7x_{B_2} = 44$ hits the left boundary at:

$$ 
(x_{A_1}, x_{A_2}) = (0,54/7) ~~~ (x_{B_1}, x_{B_2}) = (14, 2/7)
$$

I don't understand how these intercepts have been calculated. 

I've used the MRS and endowment point to calculate the $(x_{A_1}, x_{A_2})$
intercepts correctly:

$$x_{A_2} = -\frac{2}{5}x_{A_1} + \frac{38}{5}$$

But get an incorrect $(x_{B_1}, x_{B_2})$ intercept:

$$
x_{A_1} = 14, ~~ x_{A_2} = 2 
$$

Please can someone enlighten me? No worries if you can't.

Alex
In an edgeworth box model the total amount of goods must be constant. In your problem this translates to

\begin{eqnarray}
x_{A_1} + x_{B_1} &amp;=&amp; \omega_{A_1} + \omega_{B_1} = 10 + 4 = 14 \tag{1}\\
x_{A_2} + x_{B_2} &amp;=&amp; \omega_{A_2} + \omega_{B_2} = 6 + 2 = 8 \tag{2}\\
\end{eqnarray}

So let's consider the first intercept

$$
(x_{A_1}, x_{A_2}) = (0, 38/5) \tag{3}
$$

which you properly derived. If you replace these values in Eq. (1) you get

$$
0 + x_{B_1} = 14 ~~~\Rightarrow~~~ x_{B_1} = 14 \tag{4}
$$

and if you replace in Eq. (2) you get

$$
\frac{38}{5} + x_{B_2} = 8 ~~~\Rightarrow~~~ x_{B_2} = \frac{2}{5} \tag{5}
$$

So there you have it

$$
(x_{A_1}, x_{A_2}) = (0,38/5), ~~~(x_{B_1}, x_{B_2}) = (14,2/5)
$$

You can do the same for the other intercept, I will leave that for you to complete