1. **Stating the problem:** We want to solve simultaneous equations using the elimination method. For example, solve the system: $$\begin{cases} 2x + 3y = 8 \\ 4x - y = 2 \end{cases}$$ 2. **Formula and rules:** The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. 3. **Step 1: Make coefficients of one variable equal:** Multiply the second equation by 3 to match the coefficient of $y$ in the first equation: $$\begin{cases} 2x + 3y = 8 \\ 3 \times (4x - y) = 3 \times 2 \Rightarrow 12x - 3y = 6 \end{cases}$$ 4. **Step 2: Add the two equations to eliminate $y$:** $$\begin{aligned} &(2x + 3y) + (12x - 3y) = 8 + 6 \\ &2x + 3y + 12x - 3y = 14 \\ &(2x + 12x) + (3y - 3y) = 14 \\ &14x + \cancel{0} = 14 \\ &14x = 14 \end{aligned}$$ 5. **Step 3: Solve for $x$:** $$x = \frac{14}{14} = 1$$ 6. **Step 4: Substitute $x=1$ into one original equation to find $y$:** Using $2x + 3y = 8$: $$2(1) + 3y = 8 \Rightarrow 2 + 3y = 8$$ 7. **Step 5: Solve for $y$:** $$3y = 8 - 2 = 6 \Rightarrow y = \frac{6}{3} = 2$$ **Final answer:** $x=1$, $y=2$