8. \( y+11=2 x \) \( 3 y-6 x=-33 \)

To solve the system of equations: \[ \begin{cases} y + 11 = 2x \quad \text{(1)} \\ 3y - 6x = -33 \quad \text{(2)} \end{cases} \] we can use the **substitution method**. Here's a step-by-step solution: ### Step 1: Solve Equation (1) for \( y \) Starting with equation (1): \[ y + 11 = 2x \] Subtract 11 from both sides to isolate \( y \): \[ y = 2x - 11 \] ### Step 2: Substitute \( y \) into Equation (2) Now, substitute \( y = 2x - 11 \) into equation (2): \[ 3y - 6x = -33 \] \[ 3(2x - 11) - 6x = -33 \] ### Step 3: Simplify and Solve for \( x \) Expand the equation: \[ 6x - 33 - 6x = -33 \] Combine like terms: \[ 0x - 33 = -33 \] \[ -33 = -33 \] ### Step 4: Interpret the Result The equation simplifies to a **true statement** (\(-33 = -33\)) without involving \( x \) or \( y \). This indicates that both original equations are **dependent** and represent the **same line** in the coordinate plane. ### Conclusion Since both equations describe the same relationship between \( x \) and \( y \), there are **infinitely many solutions** to the system. These solutions are all the points that lie on the line defined by \( y = 2x - 11 \). **Solution Set:** \[ \boxed{\left\{\, (x, y) \mid y = 2x - 11 \right\}} \] This means any real number \( x \) will produce a corresponding \( y \) that satisfies both equations.