1. **State the problem:** Solve the system of equations using the elimination method: $$4x - 2y = 14$$ $$-10x + 7y = -25$$ 2. **Goal:** Eliminate one variable by making the coefficients of either $x$ or $y$ opposites. 3. **Eliminate $y$:** Multiply the first equation by 7 and the second equation by 2 to align the $y$ coefficients: $$7(4x - 2y) = 7(14) \Rightarrow 28x - 14y = 98$$ $$2(-10x + 7y) = 2(-25) \Rightarrow -20x + 14y = -50$$ 4. **Add the two equations:** $$ (28x - 14y) + (-20x + 14y) = 98 + (-50) $$ $$ 28x - 14y - 20x + 14y = 48 $$ $$ (28x - 20x) + (-14y + 14y) = 48 $$ $$ 8x + 0 = 48 $$ $$ 8x = 48 $$ 5. **Solve for $x$:** $$ x = \frac{48}{8} $$ $$ x = 6 $$ 6. **Substitute $x=6$ into the first original equation to find $y$:** $$ 4(6) - 2y = 14 $$ $$ 24 - 2y = 14 $$ 7. **Isolate $y$:** $$ -2y = 14 - 24 $$ $$ -2y = -10 $$ 8. **Divide both sides by -2:** $$ y = \frac{-10}{-2} $$ $$ y = 5 $$ **Final answer:** $$ x = 6, \quad y = 5 $$