1. **State the problem:** Solve a system of linear equations using the elimination method. 2. **Explain the elimination method:** The elimination method involves adding or subtracting the equations to eliminate one variable, making it easier to solve for the other. 3. **General formula:** For two equations: $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$ Multiply one or both equations by constants so that the coefficients of one variable are opposites. 4. **Example:** Suppose the system is: $$2x + 3y = 8$$ $$4x - 3y = 2$$ Add the two equations to eliminate $y$: $$ (2x + 3y) + (4x - 3y) = 8 + 2 $$ $$ 2x + 4x + 3y - 3y = 10 $$ $$ 6x + \cancel{3y - 3y} = 10 $$ $$ 6x = 10 $$ 5. **Solve for $x$:** $$ x = \frac{10}{6} = \frac{5}{3} $$ 6. **Substitute $x$ back into one original equation to find $y$:** Using $2x + 3y = 8$: $$ 2\left(\frac{5}{3}\right) + 3y = 8 $$ $$ \frac{10}{3} + 3y = 8 $$ $$ 3y = 8 - \frac{10}{3} = \frac{24}{3} - \frac{10}{3} = \frac{14}{3} $$ $$ y = \frac{14}{3} \times \frac{1}{3} = \frac{14}{9} $$ 7. **Final answer:** $$ x = \frac{5}{3}, \quad y = \frac{14}{9} $$ This completes the solution using the elimination method.