Hi Friends
and Champs!
Before to
start next exercise, we need to know some basic terms:
Consistent
pair of linear equations:
A pair of linear equations in two variables which has a solution.
Dependent
pair of linear equations:
A pair of linear equations which has infinite many solutions.
Inconsistent
pair of linear equations:
A pair of linear equations which has no solution.
Linear
General equation of a line:
ax+by+c=0
let we
have two linear equations in two variables:
a1 x
+ b1 y + c1 = 0 -------(1)
a2 x
+ b2 y + c2 = 0--------(2)
comparing
both equation (1) and (2)
If
a1/a2 = b1/b2=c1/c2
Lines are Coincident with infinite solution.
If
a1/a2 ≠ b1/b2
Lines are intersecting with
exactly one solution.
If
a1/a2 = b1/b2 ≠ c1/c2
Lines are parallel with no solution.
Exercise- 3.1
1.
Form the pair of linear equations in the following problems, and find their
solutions graphically.
(i)
10 students of Class X took part in a Mathematics quiz. If the number of girls
is 4 more than the number of boys, find the number of boys and girls who took
part in the quiz.
(ii)
5 pencils and 7 pens together cost ` 50, whereas 7 pencils and 5 pens together
cost ` 46. Find the cost of one pencil and that of one pen.
Solution:
(1)
Let there are "x" no of boys and "y" no of girls,
then x + y = 10 ------------(1)
and y - x = 4
=> x - y = - 4 --------------(2)
From
equation (1) y = 10 - x and from equation (2) y = x+4
let's make
table for both equations:
So number
of boys are = 3
and number
of girls are =7
(ii) 5
pencils and 7 pens together cost ` 50, whereas 7 pencils and 5 pens together
cost ` 46. Find the cost of one pencil and that of one pen.
Solution:
Let cost
of one pencil = x
and cost
of one pen = y
=> 5x +
7y = 50
=> y =
(50 - 5x) / 7 --------------(1)
and 7x +
5y = 46
=> y =
( 46 - 7x ) / 5 ---------------(2)
let's make
table and graph for both equations.
both lines are intersecting at y = 5 and x = 3
so cost of
one pen is 5
and cost
of one pencil is 3.
2. On
comparing the ratios find out a1/a2 , b1/b2
,c1/c2 whether the lines representing
the following pairs of linear equations intersect at a point, are parallel
or coincident:
(i)
5x – 4y + 8 = 0 7x + 6y – 9 = 0
(ii) 9x
+ 3y + 12 = 0 18x + 6y + 24 = 0
(iii)
6x – 3y + 10 = 0 2x – y + 9 = 0
Solution
:
(i) 5x
– 4y + 8 = 0 7x + 6y – 9 = 0
a1/a2 =
5/7
b1/b2 =-4/6
c1/c2 =-8/-9
so a1/a2 ≠ b1/b2≠c1/c2
lines are
intersecting at exactly one point.
(ii) 9x
+ 3y + 12 = 0 18x + 6y + 24 = 0
a1/a2 =
9/18=1/2
b1/b2 =-3/6=1/2
c1/c2 =
12/24=1/2
so a1/a2 =
b1/b2= c1/c2
lines are
coincident with infinite solution.
iii) 6x
– 3y + 10 = 0 2x – y + 9 = 0
a1/a2 =
6/2=3
b1/b2 =
-3/-1=3
c1/c2 =
10/9
so a1/a2 =
b1/b2≠ c1/c2
lines are
parallel with no solution.
3. On
comparing the ratios , a1/a2 , b1/b2
,c1/c2 find out whether the following pair of
linear equations are consistent, or inconsistent.
(i) 3x
+ 2y = 5 ; 2 2x – 3y = 7 (ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii)
(3/2)x + (5/3)y =7 ;9x – 10y = 14 (iv) 5x – 3y = 11 ;– 10x + 6y =
–22
(v)
4/3x + 2y =8 2x + 3y = 12
Solution :
5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Solution :
6. Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i)
intersecting lines (ii) parallel lines (iii) coincident
lines
Solution:
(i) 3x +
2y - 7=0
(ii) 4x + 6y -10 =0
(iii) 6x + 9y - 24 =0
Solution:
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