How to solve this particular type of simultaneous equations? [closed]

Question

I am in pickle to figure out a way to input this somewhat odd system of simultaneous equations in Mathematica

The system reads

$$a_1+2 a_2-2 a_3=3 a_2-a_3=-a_6=-a_7=a_2-a_3$$ $$a_1+a_2-a_3-a_6=2 a_2-a_6=4 a_2-2 a_3=-a_6$$ $$a_1+a_2-a_3-a_7=2 a_2-a_7$$

The solution to this peculiar system is $a_2=\frac{1}{4}a_1=\frac{1}{3}a_3$, $a_6=a_7=0$.

Here is my try

Clear[a1, a2, a3, a6, a7];
Eq1=a1+2 a2-2 a3==3 a2-a3
Eq2=a1+2 a2-2 a3==-a6
Eq3=a1+2 a2-2 a3==-a7
Eq4=a1+2 a2-2 a3==a2-a3
Eq5=a1+a2-a3-a6==2 a2-a6
Eq6=a1+a2-a3-a6==4 a2-2 a3
Eq7=a1+a2-a3-a6==-a6
Eq8=a1+a2-a3-a7==2 a2-a7
Solve[{Eq1, Eq2, Eq3, Eq4, Eq5, Eq6, Eq7, Eq8}, {a1, a2, a3, a6,a7}]

gives me a trivial solution.

Is there a way around?

Answer 1

Let's move the rhs of the equations to the lhs:

eqs = (#[[1]] - #[[2]]) & /@ {Eq1, Eq2, Eq3, Eq4, Eq5, Eq6, Eq7, Eq8}

{a1 - a2 - a3, a1 + 2 a2 - 2 a3 + a6, a1 + 2 a2 - 2 a3 + a7, a1 + a2 - a3, a1 - a2 - a3, a1 - 3 a2 + a3 - a6, a1 + a2 - a3, a1 - a2 - a3}

There are indeed five variables:

Variables @ eqs

{a1, a2, a3, a6, a7}

Construct the coefficient matrix:

mat = Part[#, 2] & @ Normal @ CoefficientArrays[#, Variables @ eqs] & /@ eqs;

MatrixForm @ mat

Dimensions @ mat

{8, 5}

There are $M=8$ equations and $N=5$ variables (which is called an overdetermined system), and the system of linear equations is homogeneous. There is a non-trivial solution only if the number of independent equations (i.e., rank of matrix mat) is $\leq N-1=5-1=4$:

MatrixRank @ mat

5

so the trivial solution is the only solution of this system.