# 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

- 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. - Draw the graphs of following lines:

(i) x = 0

(ii) y = 0

(iii) x = 1

(iv) x = -2

(v) y = 2

(vi) y = -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 - 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 - If P = {(x, y), 2x -y +3 = 0} and Q = {(x, y), x-2y+1=0}, find PQ.
- 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

**4.**(i)(-2, 5) (ii) (1, 1) (iii) (2, -1)

**5.**

**6.**28 square units