To solve the system of linear equations:

Step 1: Write the equations

We have:

1. (2x - 3y = 5)
2. (4x + 5y = 1)

Step 2: Eliminate (x)

Multiply equation 1 by 2 to make the coefficient of (x) equal to 4:
(4x - 6y = 10)

Subtract this from equation 2:
((4x + 5y) - (4x - 6y) = 1 - 10)
(11y = -9)

Solve for (y):
(y = -\frac{9}{11})

Step 3: Find (x)

Substitute (y = -\frac{9}{11}) into equation 1:
(2x - 3\left(-\frac{9}{11}\right) = 5)
(2x + \frac{27}{11} = 5)
(2x = 5 - \frac{27}{11} = \frac{55 - 27}{11} = \frac{28}{11})

Solve for (x):
(x = \frac{14}{11})

Answer: (x = \frac{14}{11}), (y = -\frac{9}{11})

(\boxed{x=\dfrac{14}{11}, y=-\dfrac{9}{11}})

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