1. **State the problem:** Solve the system of equations by elimination. For example, consider the system: $$\begin{cases} 2x + 3y = 8 \\ 4x - y = 2 \end{cases}$$ 2. **Explain the elimination method:** The goal is to eliminate one variable by adding or subtracting the equations after multiplying them by suitable numbers. 3. **Multiply equations to align coefficients:** Multiply the second equation by 3 to align the coefficients of $y$: $$\begin{cases} 2x + 3y = 8 \\ 3(4x - y) = 3(2) \Rightarrow 12x - 3y = 6 \end{cases}$$ 4. **Add the equations to eliminate $y$:** $$ (2x + 3y) + (12x - 3y) = 8 + 6 $$ $$ 2x + 3y + 12x - 3y = 14 $$ $$ (2x + 12x) + (3y - 3y) = 14 $$ $$ 14x + \cancel{0} = 14 $$ $$ 14x = 14 $$ 5. **Solve for $x$:** $$ x = \frac{14}{14} $$ $$ x = 1 $$ 6. **Substitute $x=1$ into one original equation to find $y$:** Using $2x + 3y = 8$: $$ 2(1) + 3y = 8 $$ $$ 2 + 3y = 8 $$ $$ 3y = 8 - 2 $$ $$ 3y = 6 $$ $$ y = \frac{6}{3} $$ $$ y = 2 $$ 7. **Final solution:** $$ (x, y) = (1, 2) $$ This means the system's solution is $x=1$ and $y=2$.