To solve the given system of equations using the substitution method:

Given system of equations:

1. $3x + 2y = 11$
2. $x - 2 = -4y$

Step 1: Solve the second equation for $x$:

$x - 2 = -4y$ $x = -4y + 2$

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

$3(-4y + 2) + 2y = 11$ Distribute $3$: $-12y + 6 + 2y = 11$ Combine like terms: $-10y + 6 = 11$ Subtract $6$ from both sides: $-10y = 5$ Solve for $y$: $y = -\frac{1}{2}$

Step 3: Substitute $y = -\frac{1}{2}$ back into $x = -4y + 2$:

$x = -4\left(-\frac{1}{2}\right) + 2$ $x = 2 + 2 = 4$

Solution:

The solution is: $\boxed{(4, -\frac{1}{2})}$

This matches the option: $\text{The only solution is } \mathbf{(4, -\frac{1}{2})}.$

Would you like a deeper explanation of the substitution process or have further questions?

Related Questions:

1. How does substitution compare to elimination when solving systems of equations?
2. Can this system of equations be solved graphically, and what does the solution represent?
3. What are real-world applications of solving systems of equations?
4. How can substitution be applied to nonlinear systems of equations?
5. What happens if the equations are dependent or inconsistent?

Tip:

Always check your solution by substituting the values back into both original equations to ensure correctness!