Simultaneous Elimination 003308

1. **State the problem:** We want to solve simultaneous 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 form:** Suppose we have two equations: $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$ 4. **Step 1: Make coefficients of one variable equal:** Multiply one or both equations by suitable numbers so that the coefficients of either $x$ or $y$ are the same (or opposites). 5. **Step 2: Add or subtract equations:** Add or subtract the equations to eliminate one variable. 6. **Step 3: Solve for the remaining variable:** After elimination, solve the resulting single-variable equation. 7. **Step 4: Substitute back:** Substitute the found value into one of the original equations to find the other variable. 8. **Example:** Solve $$2x + 3y = 8$$ $$4x - 3y = 2$$ - Add the two equations: $$ (2x + 3y) + (4x - 3y) = 8 + 2 $$ $$ 6x + \cancel{3y - 3y} = 10 $$ $$ 6x = 10 $$ $$ x = \frac{10}{6} = \frac{5}{3} $$ - Substitute $x=\frac{5}{3}$ into the first equation: $$ 2\left(\frac{5}{3}\right) + 3y = 8 $$ $$ \frac{10}{3} + 3y = 8 $$ $$ 3y = 8 - \frac{10}{3} = \frac{24}{3} - \frac{10}{3} = \frac{14}{3} $$ $$ y = \frac{14}{3} \div 3 = \frac{14}{9} $$ 9. **Final answer:** $$ x = \frac{5}{3}, \quad y = \frac{14}{9} $$ This completes the elimination method solution.