Simultaneous Equations Ff847D

1. **Problem Statement:** We want to learn how to solve simultaneous equations, which are sets of equations with multiple variables that are solved together. 2. **Formula and Rules:** Simultaneous equations can be solved using substitution, elimination, or graphing methods. The goal is to find values of variables that satisfy all equations simultaneously. 3. **Example:** Consider the system: $$\begin{cases} 2x + 3y = 6 \\ x - y = 1 \end{cases}$$ 4. **Step 1: Solve one equation for one variable.** From the second equation: $$x - y = 1 \implies x = y + 1$$ 5. **Step 2: Substitute into the other equation:** $$2(y + 1) + 3y = 6$$ 6. **Step 3: Simplify and solve for $y$:** $$2y + 2 + 3y = 6$$ $$5y + 2 = 6$$ $$5y = 6 - 2$$ $$5y = 4$$ $$y = \frac{4}{5}$$ 7. **Step 4: Substitute $y$ back to find $x$:** $$x = y + 1 = \frac{4}{5} + 1 = \frac{4}{5} + \frac{5}{5} = \frac{9}{5}$$ 8. **Answer:** The solution is: $$x = \frac{9}{5}, \quad y = \frac{4}{5}$$ This means the values $x=\frac{9}{5}$ and $y=\frac{4}{5}$ satisfy both equations simultaneously.