`star` Solution of System of Linear Inequalities in Two Variables

`\color{red } ✍️` We will now illustrate the method for solving a system of linear inequalities in `color(red)("two variables graphically")` through some examples.

Q 3109223118

`"Solve the following system of linear inequalities graphically ."`

`color(blue)(x + y ≥ 5)` ... (1)

`color(blue)(x – y ≤ 3)` ... (2)

`color(blue)(x + y ≥ 5)` ... (1)

`color(blue)(x – y ≤ 3)` ... (2)

The graph of linear equation

`x + y = 5` is drawn in Fig 6.11.

We note that solution of inequality `(1)` is represented by the shaded region above the line `color(blue)(x + y = 5)`, including the points on the line.

On the same set of axes, we draw the graph of the equation `color(blue)(x – y = 3)` as shown in Fig 6.11. Then we note that inequality (2) represents the shaded region above the line` x – y = 3`, `color(green)("including the points on the line.")`

Clearly, the double shaded region, common to the above two shaded regions is the required solution region of the given system of inequalities.

Q 3101401328

Solve the following system of inequalities graphically

`5x + 4y ≤ 40 ... (1)`

`x ≥ 2 ... (2)`

`y ≥ 3 ... (3)`

`5x + 4y ≤ 40 ... (1)`

`x ≥ 2 ... (2)`

`y ≥ 3 ... (3)`

We first draw the graph of the line `5x + 4y = 40, x = 2` and `y = 3` Then we note that the inequality (1) represents shaded region below the line `5x + 4y = 40` and inequality (2) represents the shaded region right of line `x = 2` but inequality (3) represents the shaded region above the line `y = 3.` Hence, shaded region (Fig 6.12) including all the point on the lines are also the solution of the given system of the linear inequalities. In many practical situations involving system of inequalities the variable x and y often represent quantities that cannot have negative values, for example, number of units produced, number of articles purchased, number of hours worked, etc. Clearly, in such cases, `x ≥ 0, y ≥ 0` and the solution region lies only in the first quadrant.

Q 3111501420

Solve the following system of inequalities

`8x + 3y ≤ 100 ... (1)`

`x ≥ 0 ... (2)`

`y ≥ 0 ... (3)`

`8x + 3y ≤ 100 ... (1)`

`x ≥ 0 ... (2)`

`y ≥ 0 ... (3)`

We draw the graph of the line `8x + 3y = 100` The inequality `8x + 3y ≤ 100` represents the shaded region below the line, including the points on the line `8x +3y =100` (Fig 6.13).

Since `x ≥ 0, y ≥ 0,` every point in the shaded region in the first quadrant, including the points on the line and the axes, represents the solution of the given system of inequalities.

Q 3121501421

Solve the following system of inequalities graphically

`x + 2y ≤ 8 ... (1)`

`2x + y ≤ 8 ... (2)`

`x > 0 ... (3)`

`y > 0 ... (4)`

`x + 2y ≤ 8 ... (1)`

`2x + y ≤ 8 ... (2)`

`x > 0 ... (3)`

`y > 0 ... (4)`

We draw the graphs of the lines `x + 2y = 8 and 2x + y = 8.`

The inequality (1) and (2) represent the region below the two lines, including the point on the respective lines.

Since `x ≥ 0, y ≥ 0,` every point in the shaded region in the first quadrant represent a solution of the given system of inequalities (Fig 6.14).