To solve the system of equations by substitution, we'll follow these steps:

1. Write the system of equations: $4x + 2y + 7 = 0 \quad \text{(Equation 1)}$ $-x - y = 6 \quad \text{(Equation 2)}$

2. Solve one equation for one variable:
   Let's solve Equation 2 for $x$ in terms of $y$.

   $-x - y = 6$ Add $x$ and $y$ to both sides: $-x = 6 + y$ Now multiply both sides by -1: $x = -6 - y$

3. Substitute the expression for $x$ into Equation 1: Now, substitute $x = -6 - y$ into Equation 1: $4x + 2y + 7 = 0$ Replace $x$ with $-6 - y$: $4(-6 - y) + 2y + 7 = 0$ Simplify: $-24 - 4y + 2y + 7 = 0$ Combine like terms: $-17 - 2y = 0$ Add 17 to both sides: $-2y = 17$ Divide both sides by -2: $y = -\frac{17}{2}$

4. Substitute $y = -\frac{17}{2}$ back into the expression for $x$: From step 2, we have $x = -6 - y$. Now, substitute $y = -\frac{17}{2}$ into this: $x = -6 - \left(-\frac{17}{2}\right)$ Simplify: $x = -6 + \frac{17}{2}$ Convert -6 into a fraction with denominator 2: $x = -\frac{12}{2} + \frac{17}{2}$ Combine the fractions: $x = \frac{5}{2}$

Final Solution:

$x = \frac{5}{2}, \quad y = -\frac{17}{2}$

Would you like to see more details or have any questions about the steps? Here are some related questions you can explore:

1. How can I solve systems of equations using elimination instead of substitution?
2. What happens if the system has no solution or infinite solutions?
3. How do you verify the solution to a system of equations?
4. What is the graphical interpretation of a system of linear equations?
5. How can I apply substitution to solve non-linear systems of equations?

Tip: Always double-check your substitution step to ensure you've correctly replaced all instances of the variable!