Solve this system of cquations: \( \left\{\begin{array}{l}3 x-2 y=-8 \\ y=\frac{3}{2} x-2\end{array}\right. \)

To solve the given system of equations: \[ \begin{cases} 3x - 2y = -8 \\ y = \dfrac{3}{2}x - 2 \end{cases} \] we can use the **substitution method**. Here's a step-by-step solution: ### Step 1: Substitute \( y \) from the Second Equation into the First From the second equation: \[ y = \dfrac{3}{2}x - 2 \] Substitute this expression for \( y \) into the first equation: \[ 3x - 2\left(\dfrac{3}{2}x - 2\right) = -8 \] ### Step 2: Simplify the Equation Expand the terms: \[ 3x - 2 \times \dfrac{3}{2}x + 2 \times 2 = -8 \] \[ 3x - 3x + 4 = -8 \] ### Step 3: Combine Like Terms Combine the \( x \)-terms: \[ 0x + 4 = -8 \] \[ 4 = -8 \] ### Step 4: Analyze the Result The equation \( 4 = -8 \) is a **contradiction**; it is never true. This means that there is no value of \( x \) and \( y \) that simultaneously satisfies both equations. ### Conclusion **The system of equations has no solution.** This is because the two equations represent parallel lines with the same slope but different y-intercepts, meaning they never intersect. **Final Answer:** There is no solution to the system because the equations describe parallel lines.