# Write the system first as a vector equation and then as a matrix equation.

Write the system first as a vector equation and then as a matrix equation.
$$\displaystyle{8}{x}_{{{1}}}-{x}_{{{2}}}={4}$$
$$\displaystyle{5}{x}_{{{1}}}+{4}{x}_{{{2}}}={1}$$
$$\displaystyle{x}_{{{1}}}-{3}{x}_{{{2}}}={2}$$

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Mary Ramirez
Step 1
The given system of equation is
$$\displaystyle{8}{x}_{{{1}}}-{x}_{{{2}}}={4}$$
$$\displaystyle{5}{x}_{{{1}}}+{4}{x}_{{{2}}}={1}$$
$$\displaystyle{x}_{{{1}}}-{3}{x}_{{{2}}}={2}$$.
We need to write the system as a vector equation. From the Definition the product of A and x is the linear combination of the columns of A where A is $$\displaystyle{m}\times{m}×{n}$$ matrix and $$\displaystyle{x}\in{R}$$ is
$Ax=\begin{bmatrix} a_{1}&a_{2}&\cdots &a_{n} \end{bmatrix}\begin{bmatrix} x_{1}\\ \vdots\\ x_{n} \end{bmatrix}=x_{1}a_{1}+x_{n}a_{n}\cdots+x_{n}a_{n}$,
that is
$x_{1}\begin{bmatrix} 8\\ 5\\ 1 \end{bmatrix}+x_{2}\begin{bmatrix} -1\\ 4\\ -3 \end{bmatrix}=\begin{bmatrix} 4\\ 1\\ 2 \end{bmatrix}$
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Philip O'Neill
And from the Theorem 3 the matrix equation is
$$\displaystyle{A}{x}={b}$$
where A in $$\displaystyle{m}\times{n}$$ matrix and $$\displaystyle{b}\in{R}^{{{m}}}$$. The matrix equation is
$\begin{bmatrix} 8 & -1\\ 5 & 4\\ 1 & -3 \end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}=\begin{bmatrix} 4\\ 1\\ 2 \end{bmatrix}$.