Method to solve equation to find the value of each variable Ok lets say I am given a question:
There are 17 males students and 19 female students.
out of the 17 male students 11 of them do hip hop while the other 6 do drag racing 
while in the female side  11 of them do hip hop while the other 8 drag racing.
out of the people who do hip hop 13 of them like to eat and 9 of them like to sleep 
out of the people who do drag racing  5 of them like to eat and 9 of them like to sleep 
so what is the number of males that like to hip hop and eat
below are the equation i have come up with
$$mhe+fhe+mhs+fhs+mde+fde+mds+fds = 36$$
$$mhe+mhs+mde+mds = 17$$
$$fhe+fhs+fde+fds = 19$$
$$fhe+fhs+mhe+mhs = 22$$
$$mde+mds+fde+fds = 14$$
$$mhe + fhe = 13$$
$$mhs + fhs = 9$$
$$mde +fde =5$$
$$mds+ fds = 9$$

Assuming e.g. mhe represents a single variable (i.e., not $m \times h \times e$), we can perform Gauss-Jordan Elimination to obtain the reduced row echelon form:
$$\left(\begin{array}{cccccccc|c} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 36 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 17 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 19 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 22 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 14 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 13 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 9 \\ \end{array}\right) \xrightarrow{\text{Gauss-Jordan}} \left(\begin{array}{cccccccc|c} 1 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & -6 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 19 \\ 0 & 0 & 1 & 
1 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 9 \\ 0 & 0 & 
0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}\right)$$
So there are infinitely many solutions, given by
$$
(-6+\alpha+\beta+\gamma,19-\alpha-\beta-\gamma,9-\alpha,\alpha,5-\beta,\beta,9-\gamma,\gamma)
$$
where $\alpha,\beta,\gamma$ are arbitrary.
