Step 1: Take a numeric example:
Suppose income M = Rs. 100, $P_x$ = Rs. 10, $P_y$ = Rs. 5. If the consumer buys only Y: $Y = 100/5 = 20$ units, X = 0. If the consumer buys only X: $X = 100/10 = 10$ units, Y = 0.
Step 2: Observe the trade-off directly:
Moving from the "all Y" point (0, 20) to the "all X" point (10, 0), as X purchased rises from 0 to 10, Y purchased falls from 20 to 0 — a clear inverse relationship.
Step 3: Generalise the slope:
The line joining these two intercept points has slope $\dfrac{0-20}{10-0} = -2$, which indeed equals $-P_x/P_y = -10/5 = -2$, confirming the negative slope comes directly from the ratio of prices.
Final Answer:
Because income and prices are fixed, more of X can only be bought by sacrificing Y, giving the line its negative slope of $-P_x/P_y$.