Step 1: Prepare a table of solutions for $x - 2y = 3$, i.e. $x = 3 + 2y$.

| $x$ | 3 | 5 | 1 |
|---|---|---|---|
| $y$ | 0 | 1 | -1 |

Step 2: Prepare a table of solutions for $3x - 8y = 7$, i.e. $x = \dfrac{7 + 8y}{3}$.

| $x$ | 5 | -3 | 13 |
|---|---|---|---|
| $y$ | 1 | -2 | 4 |

Step 3: Plot both sets of points on the same graph paper and draw the two straight lines.
Step 4: The two lines intersect at the point $(5, 1)$.
Step 5: Hence the solution is $$x = 5,\qquad y = 1$$
Verification :
$$x - 2y = 5 - 2(1) = 3 \ \checkmark$$
$$3x - 8y = 3(5) - 8(1) = 15 - 8 = 7 \ \checkmark$$