7.1 Rectangular Co–ordinates and Graphs:
Introduction
: we have learnt how numbers can be represented on
a number line ( it is denoted by a point which is no dimension).Line had one
dimension. Can we locate our house as a point on a line? Don’t we need to have longitude and latitude
to locate a city on a world map?
Similarly, we need to provide the row and column
position (
The word graph means ‘to paint or ‘to draw’.
By
convention we write positive numbers on right side of zero and negative
numbers on
the left side of zero on a number line(Refer lesson 1.1) In
the graph sheet, the horizontal line is conventionally called ‘x axis’.
Thus,
the line OX represents positive
numbers and the line OX^{1} (X^{1} is also called –x) represents
negative numbers Let
us draw a perpendicular line at O to x axis and extend this line both above
and below the x axis. Again, by convention we call this vertical line as ‘y axis’.
By
convention line OY represents positive numbers and the line OY^{1}(Y^{1}
is also called –y) represents
negative numbers. The x axis and y axis together are called ‘coordinate axes’^{} ^{ } The
coordinate axes divide the plane into 4 parts which we call as the ‘quadrants’ named
as Quadrant I, Quadrant II, Quadrant III, Quadrant IV in the anti clock wise direction.
On the graph sheet we note markings on both x axis and y axis at equal
distances (1cm) both to the left of O and right of O on x axis, as well as
above O and
below O on y axis at equal distances
with same unit of measurement (say 1cm). 1.
On OX it will be to the right of O at 1cm, 2cm, 3cm .,. 2.
On OX^{1 }to the left of O at 1cm, 2cm, 3cm ….. 3.
On OY above O at 1cm, 2cm, 3cm, …. 4.
On OY^{1 }below O at 1cm, 2cm, 3cm … 

Plotting of points on a
graph sheet:
On
I Quadrant, mark a point P (anywhere you like, above x axis and to the right
of y axis). From P draw lines parallel to both x axis and y axis. These lines
have to meet X axis and Y axis
some where forming a rectangle. The
distance from O to the point where the perpendicular line from P meets x axis
is
called ‘x
coordinate’ or ‘abscissa’ of
point P. The
distance from O to the point where the horizontal line from P meets y axis is called ‘y coordinate’ or
‘ordinate ’
of
point P. In
the adjoining graph xcoordinate for P is 2 units( cms) and ycoordinate for
P is
4 units(cms) and P can be represented as P(2,4), these coordinates are also called
‘rectangular
coordinates’ (because the closed figure is a rectangle) 

7.1 Problem 1: Plot the point P(3, 2) on the plane.
Solution:
Step
1: On a graph sheet mark(x^{1}) on x axis at 3cm to the right of O Step
2: On the same sheet mark (y^{1}) on y axis at 2cm above O Step
3: Construct a rectangle with sides Ox^{1 }and Oy^{1 } in Quadrant I. The
point where lines parallel to Ox^{1 }and OY^{ } meet is P (3, 2) Exercise: Plot the point T(2,
3) on the plane. Do
you notice that points P (3, 2), T (2, 3) are not one and the same? Since
no
two points have same coordinates x and y, the coordinate (x, y) is called
‘ordered
pair’. Exercise: Mark points Q(2, 4), R(2, 4), S(2, 4) on the plane. By
convention note that 1. Negative x coordinate is marked to the left
of O on x axis (i.e. on line X^{1}O) 2. Negative y coordinate is marked below the x
axis (i.e. on line OY^{1}) We
notice that: 1.
Point Q (2, 4) is in Quadrant II 2.
Point R (2, 4) is in Quadrant III 3.
Point S (2, 4) is in Quadrant IV 

Observations:
1.
The coordinates of the origin O is (0, 0). 2.
The coordinates of any point on x axis is (_{}x, 0). 3. The coordinates of any point
on y axis is (0, _{}y).


7.1.1
Drawing graph for a linear equation:
We know that linear equation is an equation
involving only linear polynomial or variable in first degree.
Let us consider the below mentioned relations. In these
relations let the other number be x and the first number be y
No. 
Relationship 
Equivalent equation 
1 
A
number is equal to other number 
y
= x 
2 
A
number is twice the other number 
y
= 2x 
3 
A
number is one more than twice the
other number 
y
= 2x+1 
4 
A
number is twice (the sum of other number increased by 2) 
y
= 2(x+2) 
5 
Difference
between the other number and the number is 3 
xy =3 or
–y = 3x(transposition) or y =
x3(multiply by 1) 
6 
Sum
of number and other number is 3 
x+y=
3 or y
= 3x(transposition) 
All the above equations are of first degree and are
of general form y = mx + c where c
is a constant. Even if the given equation is of the form ax+by+c = 0 it can always
be converted to the form y =mx+c. How ?
1 
ax+by+c
= 0 
Given. 
2 
by=
axc 
Transposition(subtract
both sides by ax+c) 
3 
y=
(a/b)x(c/b) 
Divide
both sides by b 
4 
y
=mx+z 
m=
(a/b), z= (c/b) 
7.1.1 Problem 1: Draw the graph for the equation x+y =3
Solution:
Step 1 : Convert the given equation to
the form of y = (i.e. LHS will
have only y). So we have y =
3x(transposition). Step
2 : For few values of x (though 2 is enough) get values of y and record them in a table
like:
Step
3 : Plot the points represented by
(x,y) coordinates on a graph sheet and
join them to get a straight line This line represents the equation x+y =3(or
y= x +3) Verification: How are we sure that this line represents the equation
x+y=3? For
x=0.5 find out the ycoordinate of the point on this straight line. We notice
that it is 2.5 and thus the point (0.5, 2.5) is on the line Substituting
this value in the given equation we notice that x+y =0.5+2.5 = 3. So any
point on the straight line we have just drawn, satisfies the given linear
equation. Similarly
we find that point (3,0) is a point on the drawn straight line and satisfies
the given equation. This proves that the graph represents the given equation.
Since
y= mx+c is a first degree equation and its graph represents a straight line,
we call first degree equations as linear equations. 

Note :
1. Though only two points are enough to draw a
straight line we have prepared more values of (x,y) to indicate that there are
many solutions to x+y=3
2. It is also clear from the graph that many
coordinates ( x, y) on the line x+y =3
{for example (4,1)}satisfy the given equation x+y=3
Alternate Method of drawing graph:
We
know that to draw a straight line we need only 2 points. Hence
why is it necessary to have tables of (x, y) for various values of x and y? Why
not have just 2 points with one point
on x axis and another point on y axis, so that by joining these two points we
can draw a straight line? A
point on x axis will always have its y coordinate as 0. Similarly a point on
y axis will always have its x coordinate
as 0 Definition: 1.
‘x intercept’ of a linear equation is the xcoordinate of
the point where graph crosses the x –axis( It is the distance of point from O
on xaxis). It’s
coordinate has to be (x,0). To get this value of x, substitute y=0 in the given
equation. 2. ‘y intercept’
of a linear equation is the ycoordinate of the point where graph
crosses the y –axis( It is the distance of point from O on yaxis). It’s
coordinate has to be (0, y).To get this value of y, substitute x=0 in the
given equation. Note
that intercept means cut and hence x intercept means point of cutting of x
axis and y intercept means point of cutting of y axis. Let
us plot the graph using this alternate method for the equation x+y =
3(Problem 7.1.1.1) By
substituting y=0 in the above equation we get x=3. Thus P(3,0) is the
xintercept By
substituting x=0 in the above equation we get y=3. Thus Q(0,3) is the
yintercept By joining P and Q we get the
graph for the line x+y =3.This is the same graph we have drawn in Problem 7.1.1.1 

7.1.1 Problem 2: Draw a graph for y = 2
Solution:
Step
1 : The equation can be
interpreted as ‘for any value of x, y
is always 2’ and hence can be represented as y =0x2 Step
2: For few values of x (though 2 is enough) find values of y and record them in a table
like:
Step
3 : Plot the points represented by (x,
y) coordinates on a graph sheet and join
them to get a straight line. This line represents the equation y=2. You
can verify that the point (2,2) is on this drawn line also satisfies the
equation y = 2. Note that this drawn line
is parallel to x axis 

7.1.1 Problem 3: Draw a graph for 2y = x.
This is of the form y=mx
Solution:
Step 1: Convert the given equation to the form of y
= (i.e. LHS will only have y) so y = 
(1/2)x Step 2: For 2 values of x find
values of y and record them in a table as shown below
Step
3: Plot the points represented by (x,
y) coordinates on a graph sheet and join them to get a straight line. This
line represents the equation y= (1/2)x You
can verify that the point (1,1/2) is on this drawn line and also satisfies
the equation y = 2. Note this drawn line passes through origin (0, 0). Thus if
y=mx, then the line passes through origin. 

7.1.1 Problem 4: Draw a graph for x = 3. The equation can be
interpreted as ‘for any value of y, x is always 3’ or x+3=0
Solution:
Step1: Since the equation does
not have y term, we can say 0y = x3
Step
2: Plot the points represented by (x,
y) coordinates on a graph sheet and join
them to get a straight line. This line represents the equation x=3 Note that this drawn line is parallel to y axis. 

7.1 Summary of
learning
No 
Points to remember 
1 
x
axis and y axis are called co ordinate axis 
2 
Any
point is represented by a ordered
pair(x, y) called coordinates of that point 
3 
Origin’s
co ordinates are (0,0) 
4 
Any
point on the x axis has (x,0) as its coordinate and any
point on the y axis has(0,y) as its coordinate 
5 
The
equation to a line is of the form y = mx+c(which is of first degree) and is
called linear graph 
6 
The
graph x = constant is a line parallel to y axis and the
graph y = constant is a line parallel
to x axis 
7 
The
graph y = mx is a line which passes
through Origin 
8 
x
intercept of a linear equation is the
xcoordinate of the point where
the line crosses the x –axis at (x,0) 
9 
y
intercept of a linear equation is the
ycoordinate of the point where
the line crosses the y –axis at (0,y) 