1. **Problem:** Solve the system of equations: $$\begin{cases} 4x - 3y = 20 \\ 2x - y = 5 \end{cases}$$ 2. **Formula and rules:** We can use substitution or elimination to solve systems of linear equations. Here, we use substitution. 3. **Step 1:** From the second equation, express $y$ in terms of $x$: $$2x - y = 5 \implies y = 2x - 5$$ 4. **Step 2:** Substitute $y = 2x - 5$ into the first equation: $$4x - 3(2x - 5) = 20$$ 5. **Step 3:** Simplify: $$4x - 6x + 15 = 20$$ $$\cancel{4x} - \cancel{6x} + 15 = 20$$ $$-2x + 15 = 20$$ 6. **Step 4:** Solve for $x$: $$-2x = 20 - 15$$ $$-2x = 5$$ $$x = \frac{5}{-2} = -\frac{5}{2}$$ 7. **Step 5:** Substitute $x = -\frac{5}{2}$ back into $y = 2x - 5$: $$y = 2\left(-\frac{5}{2}\right) - 5 = -5 - 5 = -10$$ 8. **Answer for system 1:** $$x = -\frac{5}{2}, \quad y = -10$$ --- 1. **Problem:** Solve the system: $$\begin{cases} 3x + 2y = 15000 \\ 2x + 4y = 18000 \end{cases}$$ 2. **Step 1:** Multiply the first equation by 2 and the second by 3 to align coefficients of $x$: $$\begin{cases} 6x + 4y = 30000 \\ 6x + 12y = 54000 \end{cases}$$ 3. **Step 2:** Subtract the first from the second: $$(6x + 12y) - (6x + 4y) = 54000 - 30000$$ $$\cancel{6x} + 12y - \cancel{6x} - 4y = 24000$$ $$8y = 24000$$ 4. **Step 3:** Solve for $y$: $$y = \frac{24000}{8} = 3000$$ 5. **Step 4:** Substitute $y=3000$ into the first original equation: $$3x + 2(3000) = 15000$$ $$3x + 6000 = 15000$$ $$3x = 9000$$ $$x = 3000$$ 6. **Answer for system 2:** $$x = 3000, \quad y = 3000$$ --- 1. **Problem:** Solve the system: $$\begin{cases} x - y = 2 \\ 7x - 2y = 1 \end{cases}$$ 2. **Step 1:** From the first equation, express $x$: $$x = y + 2$$ 3. **Step 2:** Substitute into the second equation: $$7(y + 2) - 2y = 1$$ 4. **Step 3:** Simplify: $$7y + 14 - 2y = 1$$ $$5y + 14 = 1$$ 5. **Step 4:** Solve for $y$: $$5y = 1 - 14$$ $$5y = -13$$ $$y = -\frac{13}{5}$$ 6. **Step 5:** Substitute $y$ back into $x = y + 2$: $$x = -\frac{13}{5} + 2 = -\frac{13}{5} + \frac{10}{5} = -\frac{3}{5}$$ 7. **Answer for system 3:** $$x = -\frac{3}{5}, \quad y = -\frac{13}{5}$$