# Converting Standard form to Canonical form

This is what my lecturer says on standard and canonical forms: "Two particular forms of the linear programming problem are of interest for m equations in n unknowns. These are called the standard form, i.e. minimise $$\underline{c}^T\underline{x}$$ subject to $$A\underline{x} ≥ \underline{b}$$, $$\underline{x} ≥ \underline{0}$$ and the canonical form, i.e. minimise $$\underline{c}^T\underline{x}$$ subject to $$A\underline{x} = \underline{b}$$, $$x ≥ 0$$."

The question says: Put the following LP into canonical form (do not try to solve) Minimise $$z=x_1+2x_2+3x_3$$ subject to $$3x_1+4x_3\leq5$$ $$5x_1+x_2+6x_3 \leq 7$$ $$8x_1+9x_3\geq2$$ With all vairables non-negative.

So I started off by trying to convert it to standard form and adding in the slack and surplus variables so I have $$3x_1+4x_3+s_1=5$$ $$5x_1+x_2+6x_3+s_2= 7$$ $$8x_1+9x_3-s_3=2$$

but I'm confused about what the vectors $$\underline{x}, \underline{c}, \underline{b}$$ and the matrix $$A$$ is.

I read another a question where the answer suggested that $$\underline{x}=\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \underline{c}= \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}, \underline{b}=\begin{pmatrix} 5\\ 7\\ 2 \end{pmatrix}, A= \begin{pmatrix} 3 & 0 & 4 \\ 5 & 1 & 6 \\ 8 & 0 & 9 \end{pmatrix}$$ but I don't see where the slack and surplus variables come into play.

Your $$\underline{b}$$ is correct. The rest should be: $$\underline{x} = \begin{pmatrix}x_1\\x_2\\x_3\\s_1\\s_2\\s_3 \end{pmatrix},\ \underline{c} = \begin{pmatrix}1\\2\\3\\0\\0\\0 \end{pmatrix},\ A = \begin{pmatrix}3 &0 &4 &1 &0 &0\\ 5 &1 &6 &0 &1 &0\\ 8 &0 &9 &0 &0 &-1 \end{pmatrix}$$
• oh okay thank you! I got $\underline{c}$ and $A$ before but I didn't realize you had to add $s_1$, $s_2$ and $s_3$ to $\underline{x}$ so I assumed they were wrong. – hiling99 Oct 5 '19 at 22:24