solution to homogeneous system of linear equations with coefficients over field $\mathbb{F}_2$

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Solve the following homogeneous systems of linear equations with coefficients over field $\mathbb{F}_2$:

 x1+x2+x3+x4+x5+x6+x7 = 0
 x1=0
 x2+x3+x4 =0
 x5=0

I want to output on the sage following:

x1=x5=0
x6=x7
x2+x3+x4=0

I want to solve a general homogeneous systems with coefficients over field $\mathbb{F}_2$ and output same as above.

I hope that someone can help. Thanks!