In Exercises \( 47-60 \), solve the linear system by the Gauss-Jordan elimination method. 47. \( \left\{\begin{array}{r}x+9 y=8 \\ 2 x+8 y=6\end{array}\right. \) 48. \( \left\{\begin{array}{r}\frac{1}{3} x+2 y=1 \\ -2 x-4 y=6\end{array}\right. \)

Below are the step‐by‐step Gauss–Jordan elimination procedures for each system. ––––––––––––––––––––––––––––––––––––––––––– Problem 47: Solve   x + 9y = 8   2x + 8y = 6 Step 1. Write the augmented matrix:   [ 1 9 | 8 ]   [ 2 8 | 6 ] Step 2. Eliminate x from the second row. Replace row 2 with (row 2 − 2·row 1):   Row 2: [2 − 2·1, 8 − 2·9, 6 − 2·8] = [0, −10, −10]   So the matrix becomes:     [ 1 9 | 8 ]     [ 0 −10 | −10 ] Step 3. Scale row 2 by dividing by –10 to make the pivot 1:   Row 2: [0, (–10)/(–10), (–10)/(–10)] = [0, 1, 1]   New matrix:     [ 1 9 | 8 ]     [ 0 1 | 1 ] Step 4. Eliminate y from the first row. Replace row 1 with (row 1 − 9·row 2):   Row 1: [1, 9 − 9·1, 8 − 9·1] = [1, 0, -1] The matrix is now in reduced row–echelon form:   [ 1 0 | -1 ]   [ 0 1 | 1 ] Solution:   x = -1, y = 1 ––––––––––––––––––––––––––––––––––––––––––– Problem 48: Solve   (1/3)x + 2y = 1   –2x – 4y = 6 Step 1. Write the augmented matrix:   [ 1/3  2 | 1 ]   [ -2  –4 | 6 ] It is often easier to work with whole numbers, so multiply the first row by 3 (this is an allowed row-equivalent operation):   Row 1 × 3: [1, 6, 3]   Now the system becomes:     [ 1 6 | 3 ]     [ -2 –4 | 6 ] Step 2. Eliminate x from row 2. Multiply row 1 by 2 and add to row 2:   2·Row 1: [2, 12, 6]   Now add to row 2:     [ -2 + 2, –4 + 12, 6 + 6 ] = [0, 8, 12]   The updated matrix is:     [ 1 6 | 3 ]     [ 0 8 | 12 ] Step 3. Scale row 2 by dividing by 8:   Row 2: [0, 8/8, 12/8] = [0, 1, 3/2]   Matrix now:     [ 1 6 | 3 ]     [ 0 1 | 3/2 ] Step 4. Eliminate y from row 1. Replace row 1 with (row 1 − 6·row 2):   Row 1: [1, 6 – 6·1, 3 – 6·(3/2)]     = [1, 0, 3 – 9]     = [1, 0, -6] The matrix in reduced row–echelon form is:   [ 1 0 | -6 ]   [ 0 1 | 3/2 ] Solution:   x = -6, y = 3/2 ––––––––––––––––––––––––––––––––––––––––––– Final Answers: 47. x = -1, y = 1 48. x = -6, y = 3/2