Step 1: Assume Initial Unit Costs

Let the cost of one TV be $100$. Since the washing machine is $40\%$ less than the TV, its cost is $60$.

$\text{TV} = 100, \text{Washing Machine} = 60$

Step 2: Calculate Initial Total Cost

Find the total cost for 5 washing machines and 7 TVs using the assumed values.

$\text{Initial Total} = 5 \times 60 + 7 \times 100 = 300 + 700 = 1000$

Step 3: Calculate New Individual Costs

The washing machine cost increases by $64\%$ ($60 \times 1.64 = 98.4$) and the TV cost decreases by $20\%$ ($100 \times 0.80 = 80$).

$C_{\text{WM new}} = 98.4, \quad C_{\text{TV new}} = 80$

Step 4: Calculate New Total Cost

Compute the total cost after the price modifications for 5 washing machines and 7 TVs.

$\text{New Total} = 5 \times 98.4 + 7 \times 80 = 492 + 560 = 1052$

Step 5: Determine Percentage Change

Calculate the net percentage increase from the original total cost of 1000 units.

$\text{Percentage Change} = \frac{1052 - 1000}{1000} \times 100 = 5.2\%$