For the given Linear programming problem, 
Maximize $Z=-x+2y,$ subject to the constraints: $x\ge 3,x+y\ge 5,x+2y\ge 6,y\ge 0$.

Given a set of $m$ linear inequalities or equations in $n$ variables, we wish to find non- negative values of these variables which will satisfy these inequalities or equations and maximise/minimise or some linear function of the variables.

2. The general form of linear programming problem(maximise or minimise):

$Z=c_1{x}_1+c_2 x_2 + ... + c_n x_n$............(i)

Subjected to 

$\left.\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+....+a_{1n}{x}_n\{\le ,=,\ge \}{b}_1\\ a_{21}{x}_1+a_{22}{x}_2+....+a_{2n}{x}_n\{\le ,=,\ge \}{b}_2\\ ...\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }....\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }....\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }...\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }...\\ a_{m1}{x}_1+a_{m2}{x}_2+....+a_{mn}{x}_n\{\le ,=,\ge \}{b}_m\end{array}\right\}.........\left(ii\right)$

and, $x_1,x_2,x_3,\dots ,x_n\ge 0$ $.......\left(iii\right)$

where,

(i) $x_1,x_2,...,x_n$ are the variables whose values we wish to determine and are called the decision variables.

(ii) The linear function $Z$ which is to be maximised or minimised is called the objective function.

(iii) The inequalities or equations in $\left(ii\right)$ are called the constraints.

(iv) The set of inequalities in $\left(iii\right)$ is known as the set of non-negativity restrictions.

(v) $b_i$ where $i=1,2,...,m$ represents the requirement or availability of the $i^{\mathrm{th}}$ constraint.

(vi) $c_j$ where $j=1,2,\dots ,n$ represents the profit or cost to the objective function of the $j^{\mathrm{th}}$ variable $x_j$

3. Solution of Linear programming problem $\left(\mathbf{LPP}\right)$:

A set of values of the decision variables which satisfy the constraints of a linear programming problem $\left(\mathrm{LPP}\right)$ is called a solution of the $\mathrm{LPP}$.

4. Feasible solution and Feasible region:

A solution of a linear programming problem which also satisfies the non-negativity restrictions of the problem is called its feasible solution. The set of all feasible solutions of a linear programming problem is called the feasible region.

5. Optimal solution:

A feasible solution which optimises (maximise or minimise) the objective function of a $\mathrm{LPP}$ is called an optimal solution of the $\mathrm{LPP}$. A linear programming problem may have many optimal solutions.

6. Extreme point theorem:

It states that if a $\mathrm{LPP}$ admits an optimal solution, then at least one of the extreme (or corner) points of the feasible region gives the optimal solution.

7. Methods to solve a linear programming problem graphically:

(i) Corner-point method:

(a) Formulate the given $\mathrm{LPP}$ in mathematical form if it is not given in mathematical form.

(b) Draw the graph of constraints by using equality sign.

(c) Define feasibility region.

(d) Find the corner points.

(e) Evaluate the value of objective function at corner points.

(f) The point where the objective function attains its optimum (maximum or minimum) value is the optimal solution of the given $\mathrm{LPP}$.

(ii) Iso-profit or Iso-cost method:

(a) Formulate the given $\mathrm{LPP}$ in mathematical form, if it is not given so.

(b) Obtain the region in $xy$$-$plane containing all points that simultaneously satisfy all constraints including non-negativity restrictions. The polygonal region so obtained is the convex set of all feasible solutions of the given $\mathrm{LPP}$ and it is also known as the feasible region.

(c) Determine the coordinates of the vertices (Corner points) of the feasible region.

(d) Select value for profit or cost, and draw isoprofit / isocost line to reveal its slope.

(e) With a maximisation problem, maintain same slope and move line up and right until it touches feasible region at one point.

(f) With a minimisation problem, maintain same slope and move line down and left until it touches feasible region at one point.

(g) Identify optimal solution as coordinates of point touched by highest possible isoprofit line or lowest possible isocost line.

(h) The point(s) so obtained determine the optimal solution(s) and the value(s) of the objective function at these point(s) give the optimal solution.