Verification
Task number: 2386
Verify, whether \(\mathbf x=(4{,}0,-3{,}0,2)^T+p(-2{,}3,2{,}1,0)^T+q(1{,}2,-1{,}0,1)^T\) is a solution of the system
\[\begin{alignat*}{11} x_1 &-& x_2 &+& 2 x_3 &+& x_4 &+& 3 x_5 &=\ & 4 \\ 2 x_1 &-& 2 x_2 &+& 5 x_3 & & &+& 7 x_5 &=\ & 7 \\ x_1 &-& x_2 &+& x_3 &+& 3 x_4 &+& x_5 &=\ & 3 \\ x_1 &-& x_2 & & &+& 5 x_4 &+& 3 x_5 &=\ & 9 \end{alignat*}\]
Resolution
It suffices to parse the solution values into the original system – either any three independent points (the solution space is a plane), or the values including parameters:
\((4-2p+q)-(3p+2q)+2(-3+2p-q)+(p)+3(2+q)=\)
\[=(4-6+6)+p(-2-3+4+1)+q(1-2-2+3)=4\]\(2(4-2p+q)-2(3p+2q)+5(-3+2p-q)+0+7(2+q)=\)
\[=(8-15+14)+p(-4-6+10)+q(2-4-5+7)=7\]\((4-2p+q)-(3p+2q)+(-3+2p-q)+3(p)+1(2+q)=\)
\[=(4-3+2)+p(-2-3+2+3)+q(1-2-1+1)=6+q\]The last equation does not hold for \(q \ne 0\).
The test showed that \(\mathbf x\) is not a solution of the system.