#### Answer

$4T_{1}-T_{2}-T_{4}=30$
$-T_{1}+4T_{2}-T_{3}=60$
$-T_{2}+4T_{3}-T_{4}=70$
$-T_{1}-T_{3}+4T_{4}=40$

#### Work Step by Step

We omit degree symbols for convenience. Since each node's temperature is the average of the temperatures at the four surrounding nodes, we simplify the following four equations to get the system above:
$T_{1}=\frac{10+20+T_{2}+T_{4}}{4}$
$T_{2}=\frac{T_{1}+20+40+T_{3}}{4}$
$T_{3}=\frac{T_{4}+T_{2}+40+30}{4}$
$T_{4}=\frac{10+T_{1}+T_{3}+30}{4}$