To test the consistency of the system of equations and solve it, we proceed as follows:

Step 1: Write down the system of equations

1. $x + 2y + 32 = 14$
2. $4x + 5y + 72 = 35$
3. $3x + 3y + 4 = 21$

Step 2: Simplify each equation

• Equation 1: $x + 2y + 32 = 14$ becomes: $x + 2y = -18$

• Equation 2: $4x + 5y + 72 = 35$ becomes: $4x + 5y = -37$

• Equation 3: $3x + 3y + 4 = 21$ becomes: $3x + 3y = 17$

Step 3: Represent as a system of linear equations

1. & \quad x + 2y = -18 \\ 2. & \quad 4x + 5y = -37 \\ 3. & \quad 3x + 3y = 17 \end{aligned}$$ ### Step 4: Check consistency and solve 1. Rewrite the equations in matrix form: $$\begin{bmatrix} 1 & 2 & -18 \\ 4 & 5 & -37 \\ 3 & 3 & 17 \end{bmatrix}$$ 2. Analyze the determinant of the coefficient matrix: \[ A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 3 & 3 \end{bmatrix} numpy as np # Coefficient matrix A = np.array([[1, 2], [4, 5], [3, 3]]) # Compute rank of A and augmented matrix to check consistency rank_A = np.linalg.matrix_rank(A) # Rank of coefficient matrix # Augmented matrix with RHS b = np.array([[-18], [-37], [17]]) augmented_matrix = np.hstack((A, b)) rank_augmented = np.linalg.matrix_rank(augmented_matrix) rank_A, rank_augmentedThe system is inconsistent because the rank of the coefficient matrix ($$\text{rank}(A) = 2$$) is less than the rank of the augmented matrix ($$\text{rank}(A|b) = 3$$). ### Conclusion: This system of equations has no solution. Would you like a deeper explanation of why inconsistency arises or help with a similar set of equations? --- Here are 5 related questions: 1. How can we determine if a system of equations is consistent or inconsistent using matrices? 2. What are the implications of the rank of a matrix in solving linear equations? 3. How do augmented matrices help in solving systems of equations? 4. What is the significance of overdetermined systems in real-world applications? 5. How do you solve a consistent system using Gaussian elimination? **Tip:** When solving systems of equations, always check for consistency first to avoid unnecessary calculations.