The given equation is:

$6x + 7y = 4x + 4y$

We are asked to find the missing value for $x$ when $y = -4$.

Step 1: Simplify the equation

First, let's move all terms involving $x$ and $y$ to one side of the equation:

$6x + 7y - 4x - 4y = 0$

Simplifying further:

$(6x - 4x) + (7y - 4y) = 0$

$2x + 3y = 0$

Step 2: Substitute $y = -4$ into the equation

Now, we substitute $y = -4$ into the equation:

$2x + 3(-4) = 0$

$2x - 12 = 0$

Step 3: Solve for $x$

Now, solve for $x$:

$2x = 12$

$x = 6$

Thus, the missing value is $x = 6$.

Final answer:

The solution to the equation is $(6, -4)$.

Let me know if you would like more details, or if you have any questions!

Here are five related questions you might want to explore:

1. How would the solution change if $y = 0$?
2. Can you express the equation $2x + 3y = 0$ in terms of $y$ as a function of $x$?
3. What is the general solution to the equation $2x + 3y = 0$?
4. How would you graph the equation $2x + 3y = 0$ in the coordinate plane?
5. How does the substitution method work for solving systems of equations?

Tip: Always simplify equations by combining like terms first—it makes solving for variables much easier.