1. **State the problem:** Solve the system of equations to find the value of $y$: $$4x + 19y = 25$$ $$7x + 14y = 30$$ 2. **Use the elimination method:** We want to eliminate one variable by making the coefficients of $x$ or $y$ the same in both equations. Multiply the first equation by 7 and the second by 4 to align the $x$ coefficients: $$7(4x + 19y) = 7(25) \Rightarrow 28x + 133y = 175$$ $$4(7x + 14y) = 4(30) \Rightarrow 28x + 56y = 120$$ 3. **Subtract the second new equation from the first:** $$\cancel{28x} + 133y - (\cancel{28x} + 56y) = 175 - 120$$ $$133y - 56y = 55$$ $$77y = 55$$ 4. **Solve for $y$:** $$y = \frac{55}{77}$$ Simplify the fraction by dividing numerator and denominator by 11: $$y = \frac{\cancel{55}^{5}}{\cancel{77}^{7}} = \frac{5}{7}$$ **Final answer:** $$y = \frac{5}{7}$$