The system of equations we have is:
[tex]\begin{gathered} -12x-13y+13z=15 \\ 7x-10y-3z=11 \\ 7x+14y+5x=-5 \end{gathered}[/tex]To write the matrix for the system of equations, we need to write only the coefficients of each variable in the matrix. Also, each column will belong to the coefficients of one variable, and each line will belong to each equation (the first line of the matrix will be for equation one, the second line for the second equation, and so on)
We start by writing the first equation into the matrix:
[tex]\begin{bmatrix}{-12} & {-13} & {13\text{ |15}} \\ {\square} & {\square} & \\ {\square} & {\square} & {\square}\end{bmatrix}[/tex]We write the coefficients of each variable x, y, and z, and then in the final column, we write the number that is after the equal symbol in this case 15 (also note that we put a line before 15 to separate the coefficients of the variables from the result of the equation).
Now we take the second equation of the system: 7x-10y-3z=11, and we write the coefficient number in the second line of the matrix:
[tex]\begin{bmatrix}{-12} & {-13} & {13\text{ |15}} \\ {7} & {-10} & {-3\text{ |11}} \\ {\square} & {\square} & {\square}\end{bmatrix}[/tex]And finally, we do the same with the third equation: 7x+14y+5z=-5, and we put the coefficients in order:
[tex]\begin{bmatrix}{-12} & {-13} & {13\text{ |15}} \\ {7} & {-10} & {-3\text{ |11}} \\ {7} & {14} & {5\text{ |-5}}\end{bmatrix}[/tex]