1 We have already studied in the previously that

1 We have already studied in the previously that

1
We have already studied in the previously that the geometrical (i.e. graphical) representation of
a linear equation in two variable is a straight line.
A pair of linear equations in two variables can be represented by two straight lines. These are
the following three possibilities of them.
(a) The two lines will intersect at a point
(b) The two lines will not intersect, i.e., they are parallel and
(c) The two lines will be coincident.
A
B
C
A pair of linear equations can be represented by both ways (i.e., algebraic and geometric ways).
2
Let us take the example that Ms. Ritika goes to International fair, Pragati Maidan with Rs. 105.
She wants to have rides on the Musical Chairs and Fair Lady Wheel. The number of times she
rides Fair Lady Wheel is half the number of rides she had on the Musical Chairs. If each ride
costs Rs.10, and the Fair Lady Wheel games cost Rs.15, she spent Rs.105. Then find the number
of times she played Fair Lady Wheel. Represent this situation algebraically and graphically
(geometrically).
The pair of Linear Equations of two variables formed is :
-x + 2y = 0
10x + 15y = 105
…… (i)
2x + 3y = 21
– x + 2y = 0
…… (i)
(i)
(ii)
x 0 2
y 0 1
If y = 0, x = 0
If y = 1; x = 4
2x + 3y = 21
(i)
(ii)
(iii)
If x = 0; y = 7
If x = 3; y = 5
If x = 6; y = 3
x 0 3 6
y 7 5 3
…… (ii)
3
The lines representing a pair of linear equations in two variables and the existence of solutions
are follows:
 In this case the lines will intersect in a single point and the pair of equations have a unique
solution.
 In this case, the lines will be parallel and the pair of equations will have no solution.
 In this case, the lines will be coincident and the pair of equations will have infinitely many
solutions.
4
5
Let us consider the previous pairs of linear equation formed in examples 1, 2 and 3 and we
should know that what kind of pair they are geometrically.
(i) – x + 2y = 0 and 2x + 3y = 21
[The lines intersect]
(ii) 3x + 5y = 21 and 6x + 10y = 42
[The lines coincide]
(iii) 2x + 3y = 10 and 4x + 6y = 12
[The lines are parallel]
If the lines represented by the equation
a1x + b1y + c1 = 0
And
a2x + b2y + c2 = 0
(i)
intersecting, then
(ii)
coincident, then
(iii)
parallel, then
and
Questions
6.
Solve graphically the following system of equations
a.
x – 3y = 3
3x – 9y = 2
b.
2x – 3y + 13 = 0
3x – 2y + 12 = 0
7.
Draw the graphs of x – y + 1 = 0 and 3x + 2y – 12 = 0. Calculate the area bounded by these lines
and x-asis.
8.
Find the values of a and b for which the following system of linear equations has infinite
number of solution:
2x – 3y = 7
(p + q)x + (2p – q)y = 3(p + q + 1)
9.
Solve graphically the following system of equations :
(a)
=
where x + y = 0 and y – x = 0
(b)
99x + 101y = 499
101x + 99y = 501
(c)
, where x = 0 and y = 0
(d)
(
X = y = 2a
)
(
)
10.
The students of a class are made to stand in rows. If 3 students are extra in a row, there would
be 1 row less. If 3 students are less in a row there would be 2 rows more. Find the number of
students in the class.
11.
The incomes of X and Y are in the ratio of 8 : 7 and their expenditures are in the ratio 19 : 16. If
each saves Rs 1250, find their incomes.
FILL IN THE BLANKS
1. If 99x + 101y = 400 and 101x + 99y = 600 then x + y = _______________.
2. The number of common solution for the system of linear equations 5x + 4y + 6 = 0 and 10x + 8y
= 12 is ______________.
3. If a : b = 7 : 3 and a + b = 20 then b = ______________.
4. If
then the value of y is _______________.
5. The system of equations 3x – ky = 5 and kx – 4y = 9 has a unique solution then value of k should
not be ________________.
6. If p + q = k, p – q = n and k > n then q is _______________. (positive/negative)
7. Two distinct natural numbers are such that the sum of are number and twice the other number
is 6. The two numbers are _______________.
8. Sum of the ages of X and Y, 12 years ago, was 48 years and sum the ages of X and Y, 12 hence
will be 96 years. Person age of X is _____________.
9. For what value of k, the following linear equations will have no solution ?
2x + 3y = 1 and (3k – 1)x + (1 – 2k)y = 2k + 3
10. If 3a + 2b + 4c = 26 and 6b + 4a + 2x = 48 then a + b + c = ________________.
VERY SHORT ANSWER TYPE QUESTION
1. Solve the following pair of linear equations by the substitution method.
2x + 3y = 19 and 5x + 4y = 37.
2. Solve the following pair of linear equations by the elimination method.
X + y = 5 and 2x – y = 4
3. Five years hence, the age of Joseph will be three times that of his son. Five years ago, Joseph’s
age was seven times that of his son. What are their present ages?
4. In a fraction, if the numerator is increased by 2 and denominator is decreased by 3, then the
fraction becomes 1. Instead, if numerator is decreased by 2 and denominator is increased by 3,
the fraction becomes 3/8. Find the fraction.
5. Solve the following pair of linear equations by using cross-multiplication method :
4x + 5y = 71 and 5x + 3y = 66
6. If we increase the length and the breadth of a rectangle each by 2 units, then the area of
rectangle by 54 square units. Find the perimeter of the rectangle.
7. The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
8. In a cyclic quadrilateral ABCD, <A = (2x + 4)0, <B = (y + 3)0, <C = (2y + 10)0, <D = (4x – 5)0. Find the
four angles.
9. The income of Baby and Lovely are in the ratio of 8 : 7 and their expenditures are in their ratio of
19 : 16. If each saves RS 1250, find their incomes.
10. The sum of the digits of a two-digit number is 15. The number obtained by reversing the order
of digits of the given number exceeds the given number by 9. Find the given number.
SHORT ANSWER TYPE QUESTION
1. If the total cost of 2 apples and 3 mangoes is Rs.22, and the cost of each is at least Rs. 3. Also an
apple costs more than a mango. Each cost represents a positive integer. Then find the cost of a
mango.
2. A two digit number is formed by either subtracting 17 from nine times the sum of the digits or
by adding 21 to 13 times the difference of the digits. Find the number.
3. Ali starts his job with certain monthly salary and earns a fixed increment every year. If his salary
was Rs. 22500 per month after 6 years of service and Rs. 30,000 per month after 11 year of
service. Find his salary after 8 years.
4. The population of a town is 25000. If in the next year, the number f males increase by 5 % and
that of females increase by 3 %, the population becomes 26 %. Find the number of males and
females in the town.
5. Solve graphically the system of linear equations:
3x – 4y + 12 = 0, 2x – y + 2 = 0 and x = 4.
6. Solve for x and y :
a(x + y) + b(x – y) = a2 – ab + b2
a(x + y) – b(x – y) = a2 + ab + b2
7. Solve for x and y :
Ax + by = 1 and bx + ay =
8. Find the values of p and q for infinite number of solutions of the following linear equations :
2x + 3y = 7, (p + q)x + (2p – q)y = 21
9. Solve for a and b : 331a + 247b = 746 and 247a + 331b = 410
10. If sum of the successors of two numbers is 42 and the difference of their predecessors is 12.
Find the numbers.
LONG ANSWER TYPE QUESTION
1. Find the solution set of
2. Sourabh purchased two varieties of ice-cream cups, vanilla and strawberry- spending a total
amount of Rs. 330. If each vanilla cup costs Rs. 25 and each strawberry cup costs Rs. 40, then in
how many different combinations could he have purchased the ice-cream cups?
3. The sum of the speed of a boat in still water and the speed of the current is 10 kmph. If the boat
takes 40% of the time to travel downstream when compared to that of upstream, then find the
difference of the speeds of the boat while travelling upstream and downstream.
4. Rohit had 13 notes in the denominations of Rs. 10, Rs. 50 and Rs. 100. The total value of the
notes with him was Rs. 830. He had more of Rs. 100 notes than that of Rs. 50 notes with him.
Find the number of Rs. 10 notes with him.
5. An examination consists of 100 questions. Two marks are awarded for every correct option. If
one mark is deducted for every wrong option and half mark is deducted for every question un
attempted, then a person scores 135. Instead, if half mark is deducted for every wrong option
and one mark is deducted for every question left, then the person scores 133. Find the number
of questions left unanswered.
ANSWER KEYS
FILL IN THE BLANKS
1. (5)
2.
(Zero)
3.
(6)
4.
(Does not exist)
5.(+23)
6.
(Positive)
7.
(4 and 1)
8.(Cannot be determined)
9.(5/13)
10.
(10)
VERY SHORT ANSWER TYPE QUESTION
1. (X = 5, Y = 3)
2.
(X = 3, Y = 2)
5. (x = 9, y = 7)
6.
(50 units)
0
0
8.(A = 70 , B = 53 , C = 1100, D = 1270)
3.
(40 YRS, 10 YRS)
4.
(8/13)
7.
9.
(990,810)
(Rs. 6000, Rs. 5250)
10.
(78)
SHORT ANSWER TYPE QUESTION
1. (Rs. 4)
2.
(73)
5.(10, 4), (6, 4) and (4/5, 18/5) 6.
8.
(p = 5, q = 1)
9.
3.
(Rs. 25,500)
(
4.
)
(a = 3, b = - 1)
LONG ANSWER TYPE QUESTION
1. (x = 1/2, y = 1)
2.
(2)
3. (6 kmph)
4.
(3)
5. (14)
(13000, 12000)
7. (
10.
)
( 26 and 14)