To solve the system of equations using the elimination method, we start with:

$-2x - 6y = -8 \quad \text{(Equation 1)}$ $-x + 6y = -22 \quad \text{(Equation 2)}$

Step 1: Add the equations

We observe that the $6y$ terms in both equations have opposite signs, so adding the two equations will eliminate the $y$ term:

$(-2x - 6y) + (-x + 6y) = (-8) + (-22)$

This simplifies to:

$-3x = -30$

Step 2: Solve for $x$

To find $x$, divide both sides by $-3$:

$x = \frac{-30}{-3} = 10$

Step 3: Substitute $x = 10$ into one of the original equations

Now, substitute $x = 10$ into Equation 2 to find $y$:

$-x + 6y = -22$ $-(10) + 6y = -22$ $-10 + 6y = -22$

Add 10 to both sides:

$6y = -12$

Now, solve for $y$ by dividing by 6:

$y = \frac{-12}{6} = -2$

Step 4: Write the solution as an ordered pair

The solution is $x = 10$ and $y = -2$. Thus, the ordered pair is:

$(10,-2)$

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. How would you solve the system using substitution?
2. What happens if both equations have the same slope?
3. How can you graphically interpret the solution to this system?
4. Can you write the same system of equations in matrix form?
5. How do you verify the solution by plugging it back into both equations?

Tip: Always check your solutions by substituting them back into the original equations.