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Here are the step-by-step solutions for each problem on the worksheet.

1. Solve the system:

$$
\begin{cases}
6x - y = -34 \\
-4x - y = 36
\end{cases}
$$

Step 1: Isolate $y$ in the first equation.
$$6x - y = -34$$
Subtract $6x$ from both sides:
$$-y = -6x - 34$$
Multiply by $-1$:
$$y = 6x + 34$$

Step 2: Substitute this expression for $y$ into the second equation.
$$-4x - (6x + 34) = 36$$
Distribute the negative sign:
$$-4x - 6x - 34 = 36$$
Combine like terms:
$$-10x - 34 = 36$$
Add 34 to both sides:
$$-10x = 70$$
Divide by -10:
$$x = -7$$

Step 3: Substitute $x = -7$ back into the equation from Step 1 to find $y$.
$$y = 6(-7) + 34$$
$$y = -42 + 34$$
$$y = -8$$

Answer: $(-7, -8)$

---

2. Solve the system:

$$
\begin{cases}
-3x - y = -2 \\
3x + 4y = -10
\end{cases}
$$

Step 1: Isolate $y$ in the first equation.
$$-3x - y = -2$$
Add $3x$ to both sides:
$$-y = 3x - 2$$
Multiply by $-1$:
$$y = -3x + 2$$

Step 2: Substitute this into the second equation.
$$3x + 4(-3x + 2) = -10$$
Distribute the 4:
$$3x - 12x + 8 = -10$$
Combine like terms:
$$-9x + 8 = -10$$
Subtract 8 from both sides:
$$-9x = -18$$
Divide by -9:
$$x = 2$$

Step 3: Substitute $x = 2$ back into the equation from Step 1.
$$y = -3(2) + 2$$
$$y = -6 + 2$$
$$y = -4$$

Answer: $(2, -4)$

---

3. Solve the system:

$$
\begin{cases}
-x - y = -2 \\
-6x - 2y = -24
\end{cases}
$$

Step 1: Isolate $y$ in the first equation.
$$-x - y = -2$$
Add $x$ to both sides:
$$-y = x - 2$$
Multiply by $-1$:
$$y = -x + 2$$

Step 2: Substitute this into the second equation.
$$-6x - 2(-x + 2) = -24$$
Distribute the -2:
$$-6x + 2x - 4 = -24$$
Combine like terms:
$$-4x - 4 = -24$$
Add 4 to both sides:
$$-4x = -20$$
Divide by -4:
$$x = 5$$

Step 3: Substitute $x = 5$ back into the equation from Step 1.
$$y = -(5) + 2$$
$$y = -3$$

Answer: $(5, -3)$

---

4. Solve the system:

$$
\begin{cases}
-4x + y = 17 \\
-3x - 4y = 8
\end{cases}
$$

Step 1: Isolate $y$ in the first equation.
$$-4x + y = 17$$
Add $4x$ to both sides:
$$y = 4x + 17$$

Step 2: Substitute this into the second equation.
$$-3x - 4(4x + 17) = 8$$
Distribute the -4:
$$-3x - 16x - 68 = 8$$
Combine like terms:
$$-19x - 68 = 8$$
Add 68 to both sides:
$$-19x = 76$$
Divide by -19:
$$x = -4$$

Step 3: Substitute $x = -4$ back into the equation from Step 1.
$$y = 4(-4) + 17$$
$$y = -16 + 17$$
$$y = 1$$

Answer: $(-4, 1)$

---

5. Solve the system:

$$
\begin{cases}
y = -x + 7 \\
y = -3x + 7
\end{cases}
$$

Step 1: Since both equations are already solved for $y$, set them equal to each other.
$$-x + 7 = -3x + 7$$

Step 2: Solve for $x$.
Add $3x$ to both sides:
$$2x + 7 = 7$$
Subtract 7 from both sides:
$$2x = 0$$
Divide by 2:
$$x = 0$$

Step 3: Substitute $x = 0$ into either original equation to find $y$.
$$y = -(0) + 7$$
$$y = 7$$

Answer: $(0, 7)$

---

6. Solve the system:

$$
\begin{cases}
y = -2x - 19 \\
y = 2x + 9
\end{cases}
$$

Step 1: Set the two expressions for $y$ equal to each other.
$$-2x - 19 = 2x + 9$$

Step 2: Solve for $x$.
Add $2x$ to both sides:
$$-19 = 4x + 9$$
Subtract 9 from both sides:
$$-28 = 4x$$
Divide by 4:
$$x = -7$$

Step 3: Substitute $x = -7$ into the second equation (it looks slightly easier).
$$y = 2(-7) + 9$$
$$y = -14 + 9$$
$$y = -5$$

Answer: $(-7, -5)$

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Final Answer:
1. (-7, -8)
2. (2, -4)
3. (5, -3)
4. (-4, 1)
5. (0, 7)
6. (-7, -5)