Solve the following systems of linear equations:

Let's solve the systems of linear equations step by step. **1. System 1:** \[\begin{cases} 3x + 7y = -5 \\ x + 4y = 7 \end{cases}\] From the second equation, we can express x as: \[x = 7 - 4y\] Substitute x in the first equation: \[3(7 - 4y) + 7y = -5\] \[21 - 12y + 7y = -5\] \[-5y = -26\] \[y = \frac{26}{5}\] Now substitute y back into the equation for x: \[x = 7 - 4(\frac{26}{5})\] \[x = 7 - \frac{104}{5}\] \[x = \frac{35 - 104}{5}\] \[x = -\frac{69}{5}\] So, the solution for the first system is: \[x = -\frac{69}{5}, y = \frac{26}{5}\] **2. System 2:** \[\begin{cases} 2x - 3y = -4 \\ 5x + y = 7 \end{cases}\] From the second equation, we can express y as: \[y = 7 - 5x\] Substitute y in the first equation: \[2x - 3(7 - 5x) = -4\] \[2x - 21 + 15x = -4\] \[17x = 17\] \[x = 1\] Now substitute x back into the equation for y: \[y = 7 - 5(1)\] \[y = 2\] So, the solution for the second system is: \[x = 1, y = 2\] **3. System 3:** \[\begin{cases} -x + y = 0 \\ 3x + y = 8 \end{cases}\] From the first equation, we can express y as: \[y = x\] Substitute y in the second equation: \[3x + x = 8\] \[4x = 8\] \[x = 2\] Since y = x: \[y = 2\] So, the solution for the third system is: \[x = 2, y = 2\] **4. System 4:** \[\begin{cases} -3x + 5y = -9 \\ x - 3y = -13 \end{cases}\] From the second equation, we can express x as: \[x = 3y - 13\] Substitute x in the first equation: \[-3(3y - 13) + 5y = -9\] \[-9y + 39 + 5y = -9\] \[-4y = -48\] \[y = 12\] Now substitute y back into the equation for x: \[x = 3(12) - 13\] \[x = 36 - 13\] \[x = 23\] So, the solution for the fourth system is: \[x = 23, y = 12\] **Final Answers:** 1. \[x = -\frac{69}{5}, y = \frac{26}{5}\] 2. \[x = 1, y = 2\] 3. \[x = 2, y = 2\] 4. \[x = 23, y = 12\] **Развёрнутый ответ на русском языке:** 1. **Первая система:** Мы выразили переменную x из второго уравнения и подставили её в первое уравнение. Затем, нашли значение y и подставили его обратно, чтобы найти x. Получили решение: $$x = -\frac{69}{5}, y = \frac{26}{5}$$. 2. **Вторая система:** Выразили y из второго уравнения, подставили в первое, нашли x. Затем, нашли значение y, подставив x обратно. Решение: $$x = 1, y = 2$$. 3. **Третья система:** Выразили y из первого уравнения (y = x) и подставили во второе. Нашли x, а затем и y. Решение: $$x = 2, y = 2$$. 4. **Четвёртая система:** Выразили x из второго уравнения, подставили в первое. Нашли y, а затем и x. Решение: $$x = 23, y = 12$$.