System Elimination 895405

1. **State the problem:** Solve the system of equations using elimination: $$\begin{cases} 2x + 3y = 8 \\ 5x + 6y = 17 \end{cases}$$ 2. **Write down the elimination method formula:** We want to eliminate one variable by making the coefficients of either $x$ or $y$ the same (or opposites) in both equations. 3. **Eliminate $y$:** Multiply the first equation by $2$ to match the coefficient of $y$ in the second equation: $$2 \times (2x + 3y) = 2 \times 8 \Rightarrow 4x + 6y = 16$$ Now the system is: $$\begin{cases} 4x + 6y = 16 \\ 5x + 6y = 17 \end{cases}$$ 4. **Subtract the second equation from the first:** $$ (4x + 6y) - (5x + 6y) = 16 - 17 $$ $$ 4x + 6y - 5x - 6y = -1 $$ $$ (4x - 5x) + (6y - 6y) = -1 $$ $$ -x + \cancel{6y} - \cancel{6y} = -1 $$ $$ -x = -1 $$ 5. **Solve for $x$:** $$ x = 1 $$ 6. **Substitute $x=1$ into one of the original equations to find $y$:** Use the first equation: $$ 2(1) + 3y = 8 $$ $$ 2 + 3y = 8 $$ $$ 3y = 8 - 2 $$ $$ 3y = 6 $$ $$ y = \frac{6}{3} $$ $$ y = 2 $$ 7. **Final answer:** The solution to the system is $$ (x, y) = (1, 2) $$