Graphing Linear inequalities in Two Variable.
A
linear equation in two variable is of the form
$ax+by+c=0$
We have already studied in
Coordinate geometry that this can be represented by a straight line in x-y plane. All the points on the straight line are the solutions of this linear equation.
we can similarly find the solution set graphically for the linear inequalities in the below form
$ax+by+c< 0$
$ax+by+c> 0$
$ax+by+c \geq 0$
$ax+by+c \leq 0 $
How to find the solution graphicaly for Linear inequalities in Two Variable.
- Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign.
$ax+by+c=0 $
- This can be done easily by Point on the x-axis( x,0) and point on the y axis ( 0,y)
- Point on x-axis given by $ax+b(0)+c=0 $ or $x=\frac {-c}{a}$ or $(\frac {-c}{a} ,0)$
- Point on y-axis given by $a(0)+by+c=0 $ or $y=\frac {-c}{b}$ or $(0,\frac {-c}{b})$
- Locate these point on cartesian plane and join them to find the line
- Use a dashed or dotted line if the problem involves a strict inequality, < or >.
- Otherwise, use a solid line to indicate that the line itself constitutes part of the solution.
- Pick a point lying in one of the half-planes determined by the line sketched in step 1 and substitute the values of x and y into the given inequality.
Use the origin whenever possible.
- If the inequality is satisfied, the graph of the inequality includes the half-plane containing the test point.
Otherwise, the solution includes the half-plane not containing the test point
Example
Determine the solution set for the inequality
$x+y > 1$
Solution
1) Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign.
i.e $x+y=1 $
2) Pick the test point as origin (0,0), and put into the inequality
$0+0 > 1$
$0> 1$
Which is false
So the solution set is other half plane of the line
How to find the solution graphicaly for pair of Linear inequalities in Two Variable.
$ax+by+c< 0$
$px+qy+c < 0$
The solution set of a system of linear inequalities in two variables x and y is the set of all points (x, y) that satisfy each inequality of the system.
Step
- Find the graphical solution for each inequality independently using the technique decsribed above
- Now determine the region in common with each solution set
Example
Find the solution of the below system of inequalities
$2x+3y >1$
$x+2y >2$
$x>1$
Solution
1) for $2x+3y >1$, Solving using the above method solution is
2)for $x+2y >2$, Solving using the above method solution is
3) for $x>1$, Solving using the above method solution is
4) Now we draw these on the single graph and can determine the common region
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