We need to start by rearranging the equations into the form: ax+by+cz=d
Equation 1:
[tex]\begin{gathered} x+3y=z+15 \\ \text{Subtract z from both sides} \\ x+3y-z=z+15-z \\ x+3y-z=15 \end{gathered}[/tex]Equation 2:
[tex]\begin{gathered} x=-3+y-z \\ \text{Subtract y from both sides} \\ x-y=-3+y-z-y \\ x-y=-3-z \\ \text{Add z to both sides} \\ x-y+z=-3-z+z \\ x-y+z=-3 \end{gathered}[/tex]Equation 3:
[tex]x+y+7z=13[/tex]Now, rewrite the system as an augmented matrix:
[tex]\begin{bmatrix}{1} & {3} & {-1|} & {15} \\ {1} & {-1} & {+1|} & {-3} \\ {1} & {1} & {+7|} & {13} \\ {} & {} & {} & {}\end{bmatrix}[/tex]1. Subtract the first row from the second row, and the result will be the second row. It is written as:
[tex]R_2-R_1\to R_2[/tex]The new matrix is:
[tex]\begin{bmatrix}{1} & {3} & {-1|} & {15} \\ {0} & {-4} & {+2|} & {-18} \\ {1} & {1} & {+7|} & {13} \\ {} & {} & {} & {}\end{bmatrix}[/tex]The next operation is:
[tex]R_3-R_1\to R_3[/tex]The matrix is:
[tex]\begin{bmatrix}{1} & {3} & {-1|} & {15} \\ {0} & {-4} & {+2|} & {-18} \\ {0} & {-2} & {+8|} & {-2} \\ {} & {} & {} & {}\end{bmatrix}[/tex]Next step:
[tex]\begin{gathered} R_3-\frac{1}{2}R_2\to R_3 \\ \begin{bmatrix}{1} & {3} & {-1|} & {15} \\ {0} & {-4} & {+2|} & {-18} \\ {0} & {0} & {+7|} & {7} \\ {} & {} & {} & {}\end{bmatrix} \end{gathered}[/tex]Next step:
[tex]\begin{gathered} \frac{1}{7}R_3\to R_3 \\ \begin{bmatrix}{1} & {3} & {-1|} & {15} \\ {0} & {-4} & {+2|} & {-18} \\ {0} & {0} & {+1|} & {1} \\ {} & {} & {} & {}\end{bmatrix} \end{gathered}[/tex]Then, we have the following system of equations:
[tex]\begin{gathered} x+3y-z=15\text{ Equation 1} \\ -4y+2z=-18\text{ Equation 2} \\ z=1\text{ Equation 3} \end{gathered}[/tex]Replace equation 3 into equation 2 and solve for y:
[tex]\begin{gathered} -4y+2\cdot1=-18 \\ -4y+2=-18 \\ -4y=-18-2 \\ -4y=-20 \\ y=\frac{-20}{-4} \\ y=5 \end{gathered}[/tex]Replace y and z into equation 1 and solve for x:
[tex]\begin{gathered} x+3\cdot5-1=15 \\ x+15-1=15 \\ x=15-15+1 \\ x=1 \end{gathered}[/tex]The solution set is { 1 , 5 , 1 }
