1. Let's start by understanding the problem: you want to rewrite given linear equations into the form $a_1x + b_1y + c_1z + d_1 = 0$ (it seems you meant a three-variable linear equation). 2. In standard form, each equation looks like $$a_1x + b_1y + c_1z + d_1 = 0,$$ where $a_1,b_1,c_1,d_1$ are constants. 3. Once you have all equations in this standard linear form, you can extract coefficients to form matrices: - The coefficient matrix $A$ will contain the $a_1, b_1, c_1$ terms from each equation as rows. - The constant vector $D$ will contain the constants $-d_1$ (moved to the right side). 4. For example, given two equations: $$2x + 3y - 4z + 5 = 0$$ $$-x + 7y + 2z - 3 = 0,$$ The coefficient matrix and constants vector are: $$A = \begin{bmatrix}2 & 3 & -4 \\ -1 & 7 & 2\end{bmatrix}, \quad D = \begin{bmatrix}-5 \\ 3\end{bmatrix}.$$ 5. This form is useful for solving systems using matrix methods like Gaussian elimination or Cramer's rule. Thus, rewriting all equations in the form $a_1x + b_1y + c_1z + d_1 = 0$ allows you to systematically express the system in matrix form for further algebraic operations.