To solve the problem, we start by using the given equations to find the ratio (\frac{5a + 2b}{3a - b}).
Step 1: Use the first equation to relate (a) and (b)
Given (2a - 3b = 0), we get:
(2a = 3b \implies a = \frac{3}{2}b)
Step 2: Let (k = \frac{5a + 2b}{3a - b})
Cross-multiplying gives:
(5a + 2b = k(3a - b))
Substitute (a = \frac{3}{2}b) into the equation:
(5\left(\frac{3}{2}b\right) + 2b = k\left(3\left(\frac{3}{2}b\right) - b\right))
Step 3: Simplify both sides
Left side: (\frac{15}{2}b + 2b = \frac{15b + 4b}{2} = \frac{19b}{2})
Right side: (k\left(\frac{9b}{2} - b\right) = k\left(\frac{7b}{2}\right))
Step 4: Solve for (k)
Equate left and right sides:
(\frac{19b}{2} = k \cdot \frac{7b}{2})
Cancel (\frac{b}{2}) (since (b \neq 0)):
(19 = 7k \implies k = \frac{19}{7})
Answer: (\boxed{\dfrac{19}{7}})


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