Linear Equations in Two Variables
3. C
1. 2.
h
3.
a
4.
(b)
Determinant Method/Cramer's Rule : If a1 x + b1 y = c1 and a2x + b2y = c2
e
c1 c2 Dx then x = = a1 b a2
r 5.
G
MULTIPLE CHOICE QUESTIONS (MCQ's)
e
2.
ni
n
3.
c
3
d
a
unique common solutions
b
two solutions
c
infinitely many solutions
d
no common solutions
2
c
y=0
Equation of Xaxis is ........ . x=0
b
x=b
d
y=a
d
1 6
If Dx = –18 and D = 3 then the value of x is ........ . a
5.
is ........ .
If the graphs of two lines intersect in a single point then the equations have ........ .
a
4.
5 2
The value of the determinants 7 4 a b 6 4
U
a
qu
(Each question carries one mark)
1.
e
b1 b2
Nature of solution depends on value of D. a) D 0 system is consistent and equations have unique solution. b) D =0 and Dx = 0 system is consistent and equations have infinite solutions. c) D = 0 and Dx 0 System is inconsistent and system have no solution.
l
c
b1 b2
So
at
c1 b1 c2 b2 xy b1 , y = = a1 b b2 a2
on
s
R
Graphical Method : Draw graphs of given linear equation by plotting minimum three suitable points. Find point of intersection of two graphs. The coordinate of this point is solution of given simultaneous equations.
ti
t
(a)
lu
p
General form of linear equation in two variables is ax + by + c = 0, where a 0 When we consider system of two linear equations in two variables, then such equations are called simultaneous equations. Solution of simultaneous equations. when the simultaneous equations are satisfied by the same pair of values off the two variables then such pair of values of the variables is known as the 'Solution' of the given simultaneous equations. Solving simultaneous equations;
6
b
–1 6
c
–1
c
–6
If x = 5 and y = 3 is the solution of 3x + ky = 3 then the value of k is ........ . a
–4
b
2
d
–2
12 c1
13.
equations have ........ . a
7.
8.
b c
no solution one solution two solution
d
infinitely many solutions
14.
(0, 0)
b
(0, 1)
c
(1, 0)
d
(1, 1) 15.
C1
11.
12.
2 3
d
c
3
d
5
16.
17.
a
(1, 6)
b
(2, 5)
c
(–1, 2)
d
(0, 4)
18.
a
parallel
b
intersecting
c
concurrent
d
coincident
19.
........ .
5 3
d
4
c
16
b
16 6 3 4
6
d
2
If the value of the determinant
m 2 –5 7
is
6
d
15
Equation of Yaxis is ........ . a
y=0
b
y=a
c
x=b
d
x=0
The graph of the equation 2x – 2y = –4 a
(0, 2)
b
(–2, 0)
c
(–2, 2)
d
(2, –2)
What is the value of D for the simultaneous a
13
b
12
c
11
d
10
What is the equation of a line parallel to (5, –6)?
a1 3x + 5y = 16 and 4x – 3y = 6, the value of a is 2
c
a
the Xaxis and passing through the point
For the given two simultaneous equations
b
If the point (3, 2) lies on the graph of the
equations 3x – y = 7; and x + 4y = 11?
If the two simultaneous equations does not have a common solution, then the graph of the two lines will be ........ .
5 –3
xy = 36 2
intersects the Xaxis is ........ .
Which point among the following lies on the graph of the equation 3x – y = 1?
a
d
So
b
1 3
x × y = 36
31, then the value of m is ........ . a b 3 9
e
1 5
c
c
If Dy = –15 and D = –5 then the value of y is ........ . a
2(x + y) = 36
lu
c
ni qu
10.
b
U
9.
2
b
ti
4x + 3y = 4 and 6x + 5y = 8, C = ........ . 2 a
1 (x + y) = 36 2
is ........ .
For the two simultaneous equations eq. 1 2 3 5
a
equation 5x + ay = 19 then the value of a
The coordinates of the point of origin are ........ . a
The perimeter of a rectangle is 36 is expressed in the mathematical equation form as ........ .
R
b1
s
a1
If a = b = c then the two simultaneous 2 2 2
MCQ's
on
6.
Unique
20.
a
y=5
b
y = –6
c
x=5
d
x = –6
If 2x + 3y = 6 and 3x + 2y = 4 then x – y is ........ . a
2
b
10
c
–2
d
6
Linear Equations in Two Variables
ALGEBRA  S.S.C. If x = 5 and y = –10 is the solution of
What is the value of k for which the simultaneous equatios 4x + 2y = 7 and kx + 3y = 5 have no solutions?
24.
kx + 4y = 5, then the value of k is ........ . c
b
5
–5
d
9
Except which value of m, the simultaneous equations mx + 5y – 11 = 0 and 14x = 15 – 7y have an unique solution? a c
23.
10
b
7
5
d
14
25.
a
4 5
b
–4 5
c
6
d
–6
The points (–2, 0) and (0, 3) lie on the
R
22.
50
graph of the equation ax – by = 12. What are the values of a and b?
One number is 5 more than seven times the other number. This is expressed in the mathematical form as ........ . b
x + 7y = 5
c
x + 5 = 7y
d
5x = 7y
c
b
4 and –6 –6 and –4
a
x = –6
b
x=6
c
y=6
d
y = –6
d
–4 and 6
What is the equation of a line parallel to the Yaxis and 6 units to the left of it?
ti
x = 7y + 5
6 and 4
lu
a
26.
a
s
a
on
21.
13
5×4–7×2=6
2.
(a)
Unique Common solution
3.
(c)
y=0
4.
(c)
D x = x = –6 D
5.
(a)
3×5+k×3=3 15 + 3k = 3 k = –4
6.
(d)
Infinitely many solutions
7.
(a)
(0, 0)
8.
(b)
C1 4 1 C2 = 8 = 2
9.
So
(b)
qu
e
1.
U
Answers
(b)
m × 7 – 2 × (–5) = 31 7m = 31 – 10 m = 3
16.
(d)
x=0
17.
(b)
(–2, 0)
18.
(a)
D =
3 –1 1 4
= 3 × 4 – (–1) × 1 = 12 + 1 = 13
ni Dy
15.
19.
(b)
y = –6
20.
(c)
2x + 3y 3x + 2y (–) (–)
= 6 = 4 (–)
y
= 2
(c)
y=
10.
(b)
(2, 5)
11.
(a)
Parallel
12.
(d)
The ratio of the coefficient of x =
13.
(b)
2(x + y) = 36
14.
(d)
5 × 3 + a × 2 = 19
k
=
2a = 19 – 15 a = 2
k
=9
D
=3
–x 21. 3 4
+
x – y = –2
(d)
x = 5, y = –10 kx + 4y = 5 k × 5 + 4(–10) = 5 5k = 45
Linear Equations in Two Variables
45 5
14 25.
a1 m b1 5 a 2 = 14 , b 2 = 7
(d)
For unique solution
b a1 1 b2 a2 m 5 14 7
m 10
23.
(a)
x = 7y + 5
24.
(c)
c b a1 = 1 1 c2 b2 a2 2 4 = 3 k
26.
(a)
k=6
12
a = –2 a = –6 When x = 0, y = 3 ax – by = 12 a(0) – b(3) = 12 b
12
= –3
b
= –4
x = –6
ti
Point (–2, 0), (0, 3) When x = –2, y = 12 ax – by = 12 a(–2) – b(0) = 12
R
mx + 5y = 11, 14x = 15 – 7y 14x + 7y = 15
s
(a)
MCQ's
on
22.
Unique
3 –1 4
a
2.
–13
b
–11
–2
c
5
d
2
origin
b
Xaxis
c
Yaxis
d
centre
no common solution
b
unique common solution
c
infinitely many solutions
d
two solutions
U
a
The point of intersection of the graph of the equation x + y = 3 with the Xaxis is ........ . (0, 3)
b
(3, 0)
c
(3, 3)
d
(0, 0)
d
–1 5
d
a1 a 2 b1 b2
If Dy = 7 and D = –35 then the value of y is ........ . a
7.
d
If the graphs of two lines coincide with each other then the equations have ........ .
a
6.
13
The point of intersection of the Xaxis and Yaxis is called ........ . a
5.
b
–4
ni qu
a
4.
c
If (a, 3) is the point lying on the graph of the equation 5x + 2y = – 4 then the value of a is ........ .
3.
11
is ........ .
So
The value of the determinant 1
e
1.
lu
Problems For Practice
5
b
1 5
c
–5
If the graphs of two lines are lines then ........ . a
a1 a 2 = b1 b2
b
a1 b 1 a2 b2
c
a1 b c = 1 1 a 2 b2 c2
Linear Equations in Two Variables
ALGEBRA  S.S.C. If the graphs of two lines are intersecting lines; then ........ . a1 b = 1 a 2 b2
a
a1 a 2 b1 b2
d
b
1
(2, 3) 3
2
2
3
5
b
(3, 2)
d
(0, 2)
c
6
d
–1
is ........ .
The value of the determinant 26 5
c
R
The value of the set
a
3 6
–4 2
5 3
2
is ........ . c
26 3
26 2
d
26 6
If the larger number (x) is divided by the smaller number (y) the quotient is 7 and the
ti
12.
b
(2, 0)
a
11.
a1 b 2 = a2 b1
The point of intersection of Yaxis and the line y = 3x + 2 is ........ . a
10.
c
s
9.
a1 b 1 a2 b2
b
on
8.
15
a
7x + y = 1
–2
So
b
4
d
7x – y = 1
c
2
d
–4
c1
For the two simultaneous equations x + 3y + 4 = 0 and 2x + 6y – 3 = 0 the value of c is ........ . 2 –4 3
a
1 2
b
c
6 3
4 3
d
If the speed of the boat in still water is x km/hr and the speed of the stream is y km/hr
qu
15.
c
x – 7y = 1
If 12x + 13y = 29 and 13x + 12y = 21 then x + y is ........ . a
14.
b
e
13.
x + 7y = 1
lu
remainder is 1. This is expressed in the mathematical equation form as ........ .
then the speed of the boat downstream is ........ . a
c
(x × y) km/hr
ni
b
(x + y) km/hr
x y km/hr
d
For the two simultaneous equations 3x + 4y = 8 and x – 2y = 5, the determinant Dx is ........ . 3 4 1 –2
a
8 5
b
U
16.
(x – y) km/hr
4 –2
c
3 8 1 5
4 8 –2 5
d
Answers 1

c
2

b
3

a
4

c
5

b
6

d
7

c
8

b
9

d
10

a
11 
d
12

b
13

c
14

a
15

c
16 
b
Linear Equations in Two Variables