Notes On Graphical Solution of a system of Linear Inequalities - CBSE Class 11 Maths

A collection of linear inequalities is called a system of linear inequalities.

Consider two linear inequalities in two variables, E1: a1x + b1y < c1,E2: a2x + b2y < c2

Let the solution sets of the equations be S1 : Solution set of E1 S2 : Solution set of E2

If S1 and S2 have some solutions in common, S3 : S1 ∩ S2 .

Consider the inequalities, x + y ≤ 5 x - y ≤ -3

Some of the many possible solutions of the inequalities, shown as ordered pairs are

S1 : {(1,0);(0,1);(-1,5);(5,-1);(2,2);(1,2);(2,1)...}

S2 : {(1,0);(0,1);(1,2);(2,1);(4,2);(2,4);(2,2);(5,3);(5,-3)...}

Consider the common elements of both the solution sets:

S : S1 ∩ S2 = {(1,2);(2,1);(1,0);(0,1)...}

These common solutions satisfy both the inequalities.

It is possible with the graphical solution of the system of inequalities to identify all the common solutions.

Solution region of the inequality x + y ≤ 5.

First, draw the graph of the equation x + y = 5 as the reference.

Consider an arbitrary point, for example, the origin.

LHS: 0 + 0 = 0

LHS: 0 < RHS: 5

Therefore, solution region is towards the origin, that is, the lower half of the plane.

In the same way, the solution region of the inequality x - y ≥ -3 is as in the figure given below.

If the solution regions of both the inequalities are overlapped, the region obtained is as given in the figure below.

The common solutions set found algebraically belong to the overlapped region on the graphs.

S : S1 ∩ S2 = {(1,2);(2,1);(1,0);(0,1)...}

Summary

A collection of linear inequalities is called a system of linear inequalities.

Consider two linear inequalities in two variables, E1: a1x + b1y < c1,E2: a2x + b2y < c2

Let the solution sets of the equations be S1 : Solution set of E1 S2 : Solution set of E2

If S1 and S2 have some solutions in common, S3 : S1 ∩ S2 .

Consider the inequalities, x + y ≤ 5 x - y ≤ -3

Some of the many possible solutions of the inequalities, shown as ordered pairs are

S1 : {(1,0);(0,1);(-1,5);(5,-1);(2,2);(1,2);(2,1)...}

S2 : {(1,0);(0,1);(1,2);(2,1);(4,2);(2,4);(2,2);(5,3);(5,-3)...}

Consider the common elements of both the solution sets:

S : S1 ∩ S2 = {(1,2);(2,1);(1,0);(0,1)...}

These common solutions satisfy both the inequalities.

It is possible with the graphical solution of the system of inequalities to identify all the common solutions.

Solution region of the inequality x + y ≤ 5.

First, draw the graph of the equation x + y = 5 as the reference.

Consider an arbitrary point, for example, the origin.

LHS: 0 + 0 = 0

LHS: 0 < RHS: 5

Therefore, solution region is towards the origin, that is, the lower half of the plane.

In the same way, the solution region of the inequality x - y ≥ -3 is as in the figure given below.

If the solution regions of both the inequalities are overlapped, the region obtained is as given in the figure below.

The common solutions set found algebraically belong to the overlapped region on the graphs.

S : S1 ∩ S2 = {(1,2);(2,1);(1,0);(0,1)...}

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