1. **Stating the problem:** We want to solve a system of equations with two variables, for example: $$\begin{cases} ax + by = c \\ dx + ey = f \end{cases}$$ 2. **Formula and rules:** The goal is to find values of $x$ and $y$ that satisfy both equations simultaneously. 3. **Methods:** Common methods include substitution, elimination, and graphing. 4. **Example using elimination:** Suppose we have: $$\begin{cases} 2x + 3y = 6 \\ 4x - y = 5 \end{cases}$$ 5. Multiply the second equation by 3 to align $y$ coefficients: $$\begin{cases} 2x + 3y = 6 \\ 3(4x - y) = 3(5) \Rightarrow 12x - 3y = 15 \end{cases}$$ 6. Add the two equations: $$ (2x + 3y) + (12x - 3y) = 6 + 15 $$ $$ 2x + 12x + 3y - 3y = 21 $$ $$ 14x + \cancel{3y} - \cancel{3y} = 21 $$ $$ 14x = 21 $$ 7. Solve for $x$: $$ x = \frac{21}{14} = \frac{3}{2} $$ 8. Substitute $x=\frac{3}{2}$ into the first equation: $$ 2\left(\frac{3}{2}\right) + 3y = 6 $$ $$ 3 + 3y = 6 $$ 9. Solve for $y$: $$ 3y = 6 - 3 = 3 $$ $$ y = \frac{3}{3} = 1 $$ 10. **Answer:** The solution to the system is: $$ (x, y) = \left(\frac{3}{2}, 1\right) $$ This means $x=1.5$ and $y=1$ satisfy both equations.