## Verification

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.