Linear Equation Graphical representation

Rectangular axes

The position of a point in a plane is fixed by selecting two axes of reference which are formed by combining two number lines at right angles so that their zeros coincide.

The horizontal number line is called x-axis and the vertical number line is called y-axis.

The point of intersection of the two number lines is called origin.

The two number lines together are called rectangular axes.

Co-ordinates

The position of a point with respect to the rectangular axes by means of a pair of numbers is called co-ordinates.

The distance OM of point P along x-axis is called x-co-ordinate or abscissa.

The distance ON of point P along y-axis is called ordinate or y-co-ordinate.

If OM=a and ON=b then position of the point P is denoted by (a, b).

Note:

Co-ordinates of the origin is (0, 0).

Co-ordinates of any point on the x-axis is (x, 0).

Co-ordinates of any point on the y-axis is (0, y).

Quadrants

The rectangular axes divide the plane into four regions called quadrant.

By convention the quadrants are numbered as I, II, III, IV in the

anticlockwise direction.

• Any point in the I quadrant will have both the co-ordinates positive.

• In the II quadrant, x-co-ordinates is negative while y-co-ordinate positive.

• In the III quadrant, x-co-ordinate as well as y-co-ordinate both are negative.

• In the IV quadrant, x-co-ordinate is positive while the y-co-ordinate is negative.

To plot the graph of a linear equation

• Re-write the given equation expressing one term in terms of the other.

Example:

Find at least 3 sets of values for the variables satisfying the equation.

2x+y=5

Express y in terms of x.

y=5-2x

Select three values of x, find corresponding values of y.

x=0, y=5-(2x0)=5

x=1, y=5-(2x1)=3

x=-1, y=5-(-2)=7

• Draw the x and y - axes. Choose a suitable scale so as to locate the selected point on the graph.

• Plot the selected points.

• Draw a straight line joining the points.

Example 1:

Plot the graph of 2x+3y=9.

Suggested answer:

Plot the graph of

2x+3y=9

2x=9-3y

(Expressing one variable in terms of the other)

Put y = 1,

x=3

Put y = -1,

x=6

Put y = 7,

x=-6

Example 2:

Plot the graph of 4x+y=4.

Suggested answer:

Plot the graph of 4x+y=4

Put y = -4,

x=2

Put y = 8,

x=-1

Put y = -8,

x=3

Example 3:

Plot the graph of 3x-2y=6.

Suggested answer:

Plot the graph of 3x-2y=6

3x-2y=6

3x=6+2y

Example 4:

Plot the graph of 5x-2y=5. Use the graph to find the area between the line and the axes.

Suggested answer:

Plot the graph of 5x-2y=5

Area between line and axes = area of D AOB

Example 5:

Plot the graph of x+3y=6. Use the graph to find

a) area between the line and axes

b) value of y when x=-6

Suggested answer:

x+3y=6

x=6-3y

y=2, x=6-3y

=6-3(2)

=6-6

x=0

y=0, x=6-3y

=6-3(0)

=6-0

x=6

y=+3, x=6-3y

=6-3(3)

=6-9

x=-3

a) Area between line and axes = area of D AOB

=6 sq.units

b) When x=-6,

x = 6 - 3y , when x = -6,

-6 = 6 - 3y

-6 - 6 = - 3y

-12 = - 3y

\ if x = -6, y = 4