Graphs

• To solve two equations graphically, we first draw the graphs of the two lines. Their point of intersection is called the solution set.

Illustrative Examples

Example

Solve the equations y = 2x +1 and x +2y +3 = 0 graphically.
Alternatively, If P = {points (x, y), y = 2x +1} and Q={points (x, y), x+2y+3=0}, find PQ.

Solution

For equation y = 2x +1, we have the following:

Table of values

x	0	1	-1
y	1	3	-1

For equation x +2y +3 = 0, we get

2y = -x -3 or y = -x/2 -3/2

Table of values

x	1	-1	-3
y	-2	-1	0

The graph is shown in above figure. We observe that these lines intersect at point (-1,-1).

Hence the solution set is P Q = {(-1,-1)}.

We can verify that point (-1, -1) satisfies both equations:
2x +1 = 2 (-1) +1 = -1 = y; x +2y +3 = -1 +2 (-1) +3 = 0.

Exercise

1. Draw the graphs of following lines:
   (i) y = 2x
   (ii) y = -3x
   (iii) y = x/2
   (iv) y = x +1
   (v) y = x -1
   (vi) y = 2x +1
   (vii) y = x/2 -2.
2. Draw the graphs of following lines:
   (i) x = 0
   (ii) y = 0
   (iii) x = 1
   (iv) x = -2
   (v) y = 2
   (vi) y = -3
3. Draw the graphs of the following lines:
   (i) x +y = 0
   (ii) x -y = 0
   (iii) 2x +3 = 0
   (iv) 2x -3y +5 = 0
   (v) 2x +2y -3 = 0
   (vi) x/3 +y/3 = 1
   (vii) x/2 -y/3 = 1
4. Draw the graphs of following pairs of lines on the same squared paper. Hence find their point of intersection.
   (i) x +y -3 = 0 and x -y +7 = 0
   (ii) x +3y -4 = 0 and 3x -y -2 = 0
   (iii) 2x +y -3 = 0 and 3x +2y -4 = 0
5. If P = {(x, y), 2x -y +3 = 0} and Q = {(x, y), x-2y+1=0}, find PQ.
6. Draw the graphs of linear equations x = -2, x = 5, y = 0 and y = 4 on the same squared paper. Hence find the area of the quadrilateral enclosed by these lines.

Answers