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Let's solve each system of linear equations using the elimination method. The goal is to eliminate one variable by adding or subtracting the equations (after multiplying if necessary), then solve for the remaining variable, and finally substitute back to find the other.

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1.

$$
\begin{align*}
-2x - 3y &= -7 \quad \text{(1)}\\
-2x - 5y &= -5 \quad \text{(2)}
\end{align*}
$$

Step 1: Subtract equation (2) from equation (1):
$$
(-2x - 3y) - (-2x - 5y) = -7 - (-5)
$$
$$
-2x - 3y + 2x + 5y = -7 + 5
$$
$$
2y = -2 \Rightarrow y = -1
$$

Step 2: Plug $ y = -1 $ into equation (1):
$$
-2x - 3(-1) = -7 \Rightarrow -2x + 3 = -7 \Rightarrow -2x = -10 \Rightarrow x = 5
$$

✔ Solution: $ (5, -1) $

---

2.

$$
\begin{align*}
-5x + 3y &= 10 \quad \text{(1)}\\
-5x - 2y &= 35 \quad \text{(2)}
\end{align*}
$$

Step 1: Subtract equation (2) from equation (1):
$$
(-5x + 3y) - (-5x - 2y) = 10 - 35
$$
$$
-5x + 3y + 5x + 2y = -25 \Rightarrow 5y = -25 \Rightarrow y = -5
$$

Step 2: Plug $ y = -5 $ into equation (1):
$$
-5x + 3(-5) = 10 \Rightarrow -5x - 15 = 10 \Rightarrow -5x = 25 \Rightarrow x = -5
$$

✔ Solution: $ (-5, -5) $

---

3.

$$
\begin{align*}
x + 2y &= -1 \quad \text{(1)}\\
x + y &= 2 \quad \text{(2)}
\end{align*}
$$

Step 1: Subtract equation (2) from equation (1):
$$
(x + 2y) - (x + y) = -1 - 2 \Rightarrow x + 2y - x - y = -3 \Rightarrow y = -3
$$

Step 2: Plug $ y = -3 $ into equation (2):
$$
x + (-3) = 2 \Rightarrow x = 5
$$

✔ Solution: $ (5, -3) $

---

4.

$$
\begin{align*}
-3x - 3y &= 12 \quad \text{(1)}\\
5x - 3y &= 20 \quad \text{(2)}
\end{align*}
$$

Step 1: Subtract equation (1) from equation (2):
$$
(5x - 3y) - (-3x - 3y) = 20 - 12
$$
$$
5x - 3y + 3x + 3y = 8 \Rightarrow 8x = 8 \Rightarrow x = 1
$$

Step 2: Plug $ x = 1 $ into equation (1):
$$
-3(1) - 3y = 12 \Rightarrow -3 - 3y = 12 \Rightarrow -3y = 15 \Rightarrow y = -5
$$

✔ Solution: $ (1, -5) $

---

5.

$$
\begin{align*}
-5x - 4y &= 8 \quad \text{(1)}\\
4x - 4y &= -28 \quad \text{(2)}
\end{align*}
$$

Step 1: Subtract equation (2) from equation (1):
$$
(-5x - 4y) - (4x - 4y) = 8 - (-28)
$$
$$
-5x - 4y - 4x + 4y = 36 \Rightarrow -9x = 36 \Rightarrow x = -4
$$

Step 2: Plug $ x = -4 $ into equation (2):
$$
4(-4) - 4y = -28 \Rightarrow -16 - 4y = -28 \Rightarrow -4y = -12 \Rightarrow y = 3
$$

✔ Solution: $ (-4, 3) $

---

6.

$$
\begin{align*}
4x + 2y &= -12 \quad \text{(1)}\\
-5x + 2y &= 6 \quad \text{(2)}
\end{align*}
$$

Step 1: Subtract equation (2) from equation (1):
$$
(4x + 2y) - (-5x + 2y) = -12 - 6
$$
$$
4x + 2y + 5x - 2y = -18 \Rightarrow 9x = -18 \Rightarrow x = -2
$$

Step 2: Plug $ x = -2 $ into equation (1):
$$
4(-2) + 2y = -12 \Rightarrow -8 + 2y = -12 \Rightarrow 2y = -4 \Rightarrow y = -2
$$

✔ Solution: $ (-2, -2) $

---

7.

$$
\begin{align*}
3x + 4y &= -12 \quad \text{(1)}\\
-3x - 2y &= 12 \quad \text{(2)}
\end{align*}
$$

Step 1: Add equations (1) and (2):
$$
(3x + 4y) + (-3x - 2y) = -12 + 12 \Rightarrow 0x + 2y = 0 \Rightarrow y = 0
$$

Step 2: Plug $ y = 0 $ into equation (1):
$$
3x + 4(0) = -12 \Rightarrow 3x = -12 \Rightarrow x = -4
$$

✔ Solution: $ (-4, 0) $

---

8.

$$
\begin{align*}
-5x + 5y &= 0 \quad \text{(1)}\\
5x + 4y &= 0 \quad \text{(2)}
\end{align*}
$$

Step 1: Add equations (1) and (2):
$$
(-5x + 5y) + (5x + 4y) = 0 + 0 \Rightarrow 0x + 9y = 0 \Rightarrow y = 0
$$

Step 2: Plug $ y = 0 $ into equation (1):
$$
-5x + 5(0) = 0 \Rightarrow -5x = 0 \Rightarrow x = 0
$$

✔ Solution: $ (0, 0) $

---

✔ Final Answers:

| Problem | Solution |
|--------|----------|
| 1 | $ (5, -1) $ |
| 2 | $ (-5, -5) $ |
| 3 | $ (5, -3) $ |
| 4 | $ (1, -5) $ |
| 5 | $ (-4, 3) $ |
| 6 | $ (-2, -2) $ |
| 7 | $ (-4, 0) $ |
| 8 | $ (0, 0) $ |

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