The matrix A given below corresponds to a system of linear equations. After Gauss-Jordan elimination is applied to A, the matrix B is obtained Write the systern of linear equations corresponding to A, and use B to give all solutions of the system of linear equations. \[ A=\left[\begin{array}{rrrr} 9 & 29 & 5 & 20 \\ 4 & 13 & 5 & -9 \\ 1 & 3 & -5 & 3 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 1 & 0 & -80 & 0 \\ 0 & 1 & 25 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \] Write the system of linear equations corresponding to the matrix A . \[ \begin{array}{l} 9 x_{1}+(29) x_{2}+(5) x_{3}=20 \\ 4 x_{1}+(13) x_{2}+(5) x_{3}=-9 \\ 1 x_{1}+(3) x_{2}+(-5) x_{3}=3 \end{array} \] Select the correct choice below and fill in any answer boxes within your choice, A. There is one solution. The solution is \( x_{1}=\square, x_{2}=\square \), and \( x_{3}=\square \). \( \square \) \( \square \) \( \square \) (Simplify your answer.) B. There are infinitely many solutions. If \( x_{3} \) is any real number, \( x_{1}=\square \) and \( x_{2}=\square \). \( \square \) (Type an expression using \( x_{3} \) as the variable.) C. There is no solution.

The system of linear equations corresponding to matrix A is given by: \[ \begin{align*} 9 x_{1} + 29 x_{2} + 5 x_{3} &= 20, \\ 4 x_{1} + 13 x_{2} + 5 x_{3} &= -9, \\ 1 x_{1} + 3 x_{2} - 5 x_{3} &= 3. \end{align*} \] Looking at matrix B after Gauss-Jordan elimination, we see that the last row indicates a free variable situation, namely \(x_3\). To express the solutions, we find from matrix B: \[ x_{1} = 80x_{3}, \quad x_{2} = -25x_{3}. \] Thus, the solution is: B. There are infinitely many solutions. If \( x_{3} \) is any real number, \( x_{1} = 80x_{3} \) and \( x_{2} = -25x_{3} \). --- Throughout history, the method of solving systems of linear equations has evolved from geometric interpretations to algebraic methods. Ancient Greeks used geometric approaches, while matrices emerged in the 19th century as mathematicians like Cayley and Sylvester formalized the concept. In real-world applications, systems of linear equations model countless scenarios from economics to engineering. For instance, in resource allocation, companies can use these systems to optimize production levels based on constraints like budget and material availability. This helps in making smart business decisions based on data-driven insights!