linear algebra exercise problem, please help me with this linear combination To express each equation of this system $$x_1-x_3=0 \qquad x_2+x_3=0$$ as a linear combination of this system $$-x_1+x_2+4x_3=0 \qquad x_1+3x_2+8x_3=0 \qquad \frac{1}{2}x_1+x_2+\frac{5}{2}x_3=0$$
but I found $\begin{pmatrix}-a+b+\frac{c}{2}\\ a+3b+c\\4a+8b+\frac{5c}{2} \end{pmatrix}= \begin{pmatrix} 1\\0\\-1\end{pmatrix}$ and  $\begin{pmatrix}-a+b+\frac{c}{2}\\ a+3b+c\\4a+8b+\frac{5c}{2} \end{pmatrix}= \begin{pmatrix} 0\\1\\1\end{pmatrix}$
are incosistent , then how can we make the above said linear combination? please help me with best regards

let's call the equations in the basis set as follows: 
$$\begin{align}
E1 &: -x_1+x_2+4x_3 &=0 \\
E2 &: x_1+3x_2+8x_3 &=0 \\
E3 &: \frac{1}{2}x_1+x_2+\frac{5}{2}x_3 &=0
\end{align}$$
Now, we want to find linear combinations of these three equations that give each of the following equations: 
$$\begin{align}
R1 &: x_1 - x_3 &=0 \\
R2 &: x_2 + x_3&=0 \\
\end{align}$$
notice that $\frac{-1}{8}E1 + \frac{3}{8} E2 = E3$ so, E3 is a linear combination of $E1, E2$. It adds nothing…it is redundant. So, we can work with the first two equations. 
After some simple algebraic manipulations, I get...
$$\frac{-3}{4}E1 + \frac{1}{4} E2 = R1$$ 
The second result, R2 is not a linear combination of the equations E2, E2 and E3. 
