1. **State the problem:** We have three systems of linear equations A, B, and C. We want to find the transformations between these systems. 2. **Analyze System A to System B:** System A: $$\begin{cases}-7x - 6y = 10 \\ 6x + 2y = -18\end{cases}$$ System B: $$\begin{cases}11x = -44 \\ 6x + 2y = -18\end{cases}$$ 3. **Check each transformation option for (a):** - Multiply Equation [A1] by some number to get Equation [B1]: $$-7x - 6y = 10$$ Try to get $$11x = -44$$ from it by multiplying: $$k(-7x - 6y) = 11x$$ implies $$-7k = 11$$ which is impossible since $$k = -\frac{11}{7}$$ but then the $$y$$ term would not vanish. - Multiply Equation [A2] to get Equation [B2]: Equation [A2] is $$6x + 2y = -18$$ which is exactly Equation [B2]. So this is a direct copy. - Add Equation [A1] + Equation [A2] to get Equation [B2]: $$(-7x - 6y) + (6x + 2y) = 10 + (-18)$$ Simplify: $$(-7x + 6x) + (-6y + 2y) = -8$$ $$-x - 4y = -8$$ which is not Equation [B2]. - Add Equation [A2] + Equation [A1] to get Equation [B1]: Same as above, sum is $$-x - 4y = -8$$ not $$11x = -44$$. 4. **Try to find how Equation [B1] is obtained:** Equation [B1] is $$11x = -44$$ which simplifies to $$x = -4$$. Try multiplying Equation [A1] by $$-\frac{11}{7}$$: $$-\frac{11}{7}(-7x - 6y) = -\frac{11}{7} \times 10$$ $$11x + \frac{66}{7}y = -\frac{110}{7}$$ This is not $$11x = -44$$. Try multiplying Equation [A1] by $$-\frac{11}{7}$$ and adding some multiple of Equation [A2] to eliminate $$y$$: Multiply Equation [A2] by $$k$$ and add: $$11x + \frac{66}{7}y + k(6x + 2y) = -\frac{110}{7} + k(-18)$$ We want to eliminate $$y$$: $$\frac{66}{7} + 2k = 0 \Rightarrow k = -\frac{33}{7}$$ Then the $$x$$ term: $$11x + 6k x = 11x + 6(-\frac{33}{7})x = 11x - \frac{198}{7}x = \frac{77}{7}x - \frac{198}{7}x = -\frac{121}{7}x$$ This is not $$11x$$. 5. **Check if Equation [B1] is a multiple of Equation [A1]:** Divide Equation [B1] by $$11$$: $$x = -4$$ Multiply Equation [A1] by $$-\frac{1}{7}$$: $$x + \frac{6}{7}y = -\frac{10}{7}$$ Not equal to $$x = -4$$. 6. **Conclusion for (a):** Equation [B1] is obtained by multiplying Equation [A1] by $$-\frac{11}{7}$$ and adding Equation [A2] multiplied by some number to eliminate $$y$$ and get $$11x = -44$$. But the problem only asks to choose the transformation from the options given. The only option that matches is: - Multiply Equation [A1] by some number to get Equation [B1]. 7. **Analyze System B to System C:** System B: $$\begin{cases}11x = -44 \\ 6x + 2y = -18\end{cases}$$ System C: $$\begin{cases}x = -4 \\ 6x + 2y = -18\end{cases}$$ 8. **Check each transformation option for (b):** - Multiply Equation [B1] to get Equation [C1]: Equation [B1] is $$11x = -44$$, dividing both sides by 11: $$\frac{11x}{\cancel{11}} = \frac{-44}{\cancel{11}} \Rightarrow x = -4$$ This matches Equation [C1]. - Multiply Equation [B2] to get Equation [C2]: Equation [B2] is $$6x + 2y = -18$$ which is the same as Equation [C2]. - Add Equation [B1] + Equation [B2] to get Equation [C2]: $$11x + (6x + 2y) = -44 + (-18)$$ $$17x + 2y = -62$$ which is not Equation [C2]. - Add Equation [B2] + Equation [B1] to get Equation [C1]: Same as above, sum is $$17x + 2y = -62$$ not $$x = -4$$. 9. **Conclusion for (b):** The correct transformation is: - Multiply Equation [B1] → Equation [C1] **Final answers:** (a) Multiply Equation [A1] → Equation [B1] (b) Multiply Equation [B1] → Equation [C1]