To solve the system of equations using the elimination method:

1. Given equations:
   x + 3y = 11 \tag{1}
   4x - 7y = 6 \tag{2}

Step 1: Eliminate one variable

We aim to eliminate $x$ or $y$. To do this, multiply the equations to make the coefficients of $x$ (or $y$) equal. Here, let's eliminate $x$.

Multiply $(1)$ by $4$ (to match the coefficient of $x$ in $(2)$):

4(x + 3y) = 4(11) \implies 4x + 12y = 44 \tag{3}

Now subtract $(2)$ from $(3)$:

$(4x + 12y) - (4x - 7y) = 44 - 6$

Simplify:

$4x - 4x + 12y + 7y = 38 \implies 19y = 38$

Solve for $y$:

$y = \frac{38}{19} = 2$

Step 2: Substitute $y = 2$ into one of the original equations

Substitute $y = 2$ into $(1)$:

$x + 3(2) = 11$

Simplify:

$x + 6 = 11 \implies x = 5$

Solution:

The solution to the system is:

$x = 5, \, y = 2$

Verification:

Substitute $x = 5, y = 2$ into $(2)$:

$4(5) - 7(2) = 20 - 14 = 6$

Both equations are satisfied.

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Would you like further clarification or details on any step?
Here are related questions to expand on this:

1. How can the elimination method be used for three-variable systems?
2. What happens if a system has no solution or infinitely many solutions?
3. Can you explain how to solve this system using substitution instead?
4. How does elimination compare to matrix methods for solving systems?
5. Could we use graphing to verify the solution here?

Tip: Always double-check your elimination by verifying the solution in both original equations.