Page 1
Q u e s t i o n : 1
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
3x + 5y = 7.5
ii
2x -
y
5
+6 = 0
iii
3y – 2x = 6
iv
4x = 5y
v
x
5
-
y
6
= 1
vi
v
2x +
v
3y = 5
S o l u t i o n :
i
3x + 5y = 7.5
This can be expressed in the form ax + by + c = 0 as 3x +5y +(-7. 5) = 0
.
ii
2x -
y
5
+6 = 0
This can be expressed in the form ax + by + c = 0 as 2x + -
1
5
y +6 = 0
iii
3y – 2x = 6
This can be expressed in the form ax + by + c = 0 as 2x +(-3y)+6 = 0
.
iv
4x = 5y
This can be expressed in the form ax + by + c = 0 as 4x -5y +0 = 0
v
x
5
-
y
6
= 1
This can be expressed in the form ax + by + c = 0 as 6x -5y = 30
vi
v
2x +
v
3y = 5
This can be expressed in the form ax + by + c = 0 as
v
2x +
v
3y +(-5) = 0
Q u e s t i o n : 2
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
x = 6
ii
3x – y = x – 1
iii
2x + 9 = 0
iv
4y = 7
v
x + y = 4
vi
x
2
-
y
3
=
1
6
+y
( )
Page 2
Q u e s t i o n : 1
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
3x + 5y = 7.5
ii
2x -
y
5
+6 = 0
iii
3y – 2x = 6
iv
4x = 5y
v
x
5
-
y
6
= 1
vi
v
2x +
v
3y = 5
S o l u t i o n :
i
3x + 5y = 7.5
This can be expressed in the form ax + by + c = 0 as 3x +5y +(-7. 5) = 0
.
ii
2x -
y
5
+6 = 0
This can be expressed in the form ax + by + c = 0 as 2x + -
1
5
y +6 = 0
iii
3y – 2x = 6
This can be expressed in the form ax + by + c = 0 as 2x +(-3y)+6 = 0
.
iv
4x = 5y
This can be expressed in the form ax + by + c = 0 as 4x -5y +0 = 0
v
x
5
-
y
6
= 1
This can be expressed in the form ax + by + c = 0 as 6x -5y = 30
vi
v
2x +
v
3y = 5
This can be expressed in the form ax + by + c = 0 as
v
2x +
v
3y +(-5) = 0
Q u e s t i o n : 2
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
x = 6
ii
3x – y = x – 1
iii
2x + 9 = 0
iv
4y = 7
v
x + y = 4
vi
x
2
-
y
3
=
1
6
+y
( )
S o l u t i o n :
i
x = 6
In the form of ax + by + c = 0 we have x +0y +(-6) = 0
where a = 1, b = 0 and c = -6
.
ii
3x – y = x – 1
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iii
2x + 9 = 0
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iv
4y = 7
In the form of ax + by + c = 0 we have 0x +4y +(-7) = 0
where a = 0, b = 4 and c = -7
v
x + y = 4
In the form of ax + by + c = 0 we have 1x +1y +(-4) = 0
where a = 1, b = 1, c = -4
vi
x
2
-
y
3
=
1
6
+y
In the form of ax + by + c = 0 we have 3x -2y = 1 +6y ? 3x -8y +(-1) = 0
where a = 3, b = -8, c = -1
.
Q u e s t i o n : 3
Check which of the following are the solutions of the equation 5x – 4y = 20.
i
4, 0
ii
0, 5
iii
-2,
5
2
iv
0, – 5
v
2,
-5
2
S o l u t i o n :
The equation given is 5x – 4y = 20.
i
4, 0
Putting the value in the given equation we have
LHS: 5(4)-4(0) = 20RHS: 20LHS = RHS
Thus, 4, 0
is a solution of the given equation.
ii
0, 5
Putting the value in the given equation we have
LHS: 5(0)-4(5) = 0 -20 = -20RHS: 20LHS ? RHS
( )
( )
Page 3
Q u e s t i o n : 1
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
3x + 5y = 7.5
ii
2x -
y
5
+6 = 0
iii
3y – 2x = 6
iv
4x = 5y
v
x
5
-
y
6
= 1
vi
v
2x +
v
3y = 5
S o l u t i o n :
i
3x + 5y = 7.5
This can be expressed in the form ax + by + c = 0 as 3x +5y +(-7. 5) = 0
.
ii
2x -
y
5
+6 = 0
This can be expressed in the form ax + by + c = 0 as 2x + -
1
5
y +6 = 0
iii
3y – 2x = 6
This can be expressed in the form ax + by + c = 0 as 2x +(-3y)+6 = 0
.
iv
4x = 5y
This can be expressed in the form ax + by + c = 0 as 4x -5y +0 = 0
v
x
5
-
y
6
= 1
This can be expressed in the form ax + by + c = 0 as 6x -5y = 30
vi
v
2x +
v
3y = 5
This can be expressed in the form ax + by + c = 0 as
v
2x +
v
3y +(-5) = 0
Q u e s t i o n : 2
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
x = 6
ii
3x – y = x – 1
iii
2x + 9 = 0
iv
4y = 7
v
x + y = 4
vi
x
2
-
y
3
=
1
6
+y
( )
S o l u t i o n :
i
x = 6
In the form of ax + by + c = 0 we have x +0y +(-6) = 0
where a = 1, b = 0 and c = -6
.
ii
3x – y = x – 1
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iii
2x + 9 = 0
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iv
4y = 7
In the form of ax + by + c = 0 we have 0x +4y +(-7) = 0
where a = 0, b = 4 and c = -7
v
x + y = 4
In the form of ax + by + c = 0 we have 1x +1y +(-4) = 0
where a = 1, b = 1, c = -4
vi
x
2
-
y
3
=
1
6
+y
In the form of ax + by + c = 0 we have 3x -2y = 1 +6y ? 3x -8y +(-1) = 0
where a = 3, b = -8, c = -1
.
Q u e s t i o n : 3
Check which of the following are the solutions of the equation 5x – 4y = 20.
i
4, 0
ii
0, 5
iii
-2,
5
2
iv
0, – 5
v
2,
-5
2
S o l u t i o n :
The equation given is 5x – 4y = 20.
i
4, 0
Putting the value in the given equation we have
LHS: 5(4)-4(0) = 20RHS: 20LHS = RHS
Thus, 4, 0
is a solution of the given equation.
ii
0, 5
Putting the value in the given equation we have
LHS: 5(0)-4(5) = 0 -20 = -20RHS: 20LHS ? RHS
( )
( )
Thus, 0, 5
is not a solution of the given equation.
iii
-2,
5
2
Putting the value in the given equation we have
LHS: 5(-2)-4
5
2
= -10 -10 = -20RHS: 20LHS ? RHS
Thus, -2,
5
2
is not a solution of the given equation.
iv
0, – 5
Putting the value in the given equation we have
LHS: 5(0)-4(-5) = 0 +20 = 20RHS: 20LHS = RHS
Thus, 0, – 5
is a solution of the given equation.
v
2,
-5
2
Putting the value in the given equation we have
LHS: 5(2)-4
-5
2
= 10 +10 = 20RHS: 20LHS = RHS
Thus, 2,
-5
2
is a solution of the given equation.
Q u e s t i o n : 4
Find five different solutions of each of the following equations:
i
2x – 3y = 6
ii
2x
5
+
3y
10
= 3
iii
3y = 4x
S o l u t i o n :
i
2x – 3y = 6
x 0 3
-
3
9
2
2
y
-
2
0
-
4
1
-2
3
ii
( )
( )
( )
( )
( )
( )
Page 4
Q u e s t i o n : 1
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
3x + 5y = 7.5
ii
2x -
y
5
+6 = 0
iii
3y – 2x = 6
iv
4x = 5y
v
x
5
-
y
6
= 1
vi
v
2x +
v
3y = 5
S o l u t i o n :
i
3x + 5y = 7.5
This can be expressed in the form ax + by + c = 0 as 3x +5y +(-7. 5) = 0
.
ii
2x -
y
5
+6 = 0
This can be expressed in the form ax + by + c = 0 as 2x + -
1
5
y +6 = 0
iii
3y – 2x = 6
This can be expressed in the form ax + by + c = 0 as 2x +(-3y)+6 = 0
.
iv
4x = 5y
This can be expressed in the form ax + by + c = 0 as 4x -5y +0 = 0
v
x
5
-
y
6
= 1
This can be expressed in the form ax + by + c = 0 as 6x -5y = 30
vi
v
2x +
v
3y = 5
This can be expressed in the form ax + by + c = 0 as
v
2x +
v
3y +(-5) = 0
Q u e s t i o n : 2
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
x = 6
ii
3x – y = x – 1
iii
2x + 9 = 0
iv
4y = 7
v
x + y = 4
vi
x
2
-
y
3
=
1
6
+y
( )
S o l u t i o n :
i
x = 6
In the form of ax + by + c = 0 we have x +0y +(-6) = 0
where a = 1, b = 0 and c = -6
.
ii
3x – y = x – 1
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iii
2x + 9 = 0
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iv
4y = 7
In the form of ax + by + c = 0 we have 0x +4y +(-7) = 0
where a = 0, b = 4 and c = -7
v
x + y = 4
In the form of ax + by + c = 0 we have 1x +1y +(-4) = 0
where a = 1, b = 1, c = -4
vi
x
2
-
y
3
=
1
6
+y
In the form of ax + by + c = 0 we have 3x -2y = 1 +6y ? 3x -8y +(-1) = 0
where a = 3, b = -8, c = -1
.
Q u e s t i o n : 3
Check which of the following are the solutions of the equation 5x – 4y = 20.
i
4, 0
ii
0, 5
iii
-2,
5
2
iv
0, – 5
v
2,
-5
2
S o l u t i o n :
The equation given is 5x – 4y = 20.
i
4, 0
Putting the value in the given equation we have
LHS: 5(4)-4(0) = 20RHS: 20LHS = RHS
Thus, 4, 0
is a solution of the given equation.
ii
0, 5
Putting the value in the given equation we have
LHS: 5(0)-4(5) = 0 -20 = -20RHS: 20LHS ? RHS
( )
( )
Thus, 0, 5
is not a solution of the given equation.
iii
-2,
5
2
Putting the value in the given equation we have
LHS: 5(-2)-4
5
2
= -10 -10 = -20RHS: 20LHS ? RHS
Thus, -2,
5
2
is not a solution of the given equation.
iv
0, – 5
Putting the value in the given equation we have
LHS: 5(0)-4(-5) = 0 +20 = 20RHS: 20LHS = RHS
Thus, 0, – 5
is a solution of the given equation.
v
2,
-5
2
Putting the value in the given equation we have
LHS: 5(2)-4
-5
2
= 10 +10 = 20RHS: 20LHS = RHS
Thus, 2,
-5
2
is a solution of the given equation.
Q u e s t i o n : 4
Find five different solutions of each of the following equations:
i
2x – 3y = 6
ii
2x
5
+
3y
10
= 3
iii
3y = 4x
S o l u t i o n :
i
2x – 3y = 6
x 0 3
-
3
9
2
2
y
-
2
0
-
4
1
-2
3
ii
( )
( )
( )
( )
( )
( )
2x
5
+
3y
10
= 3
x 0
15
2
5 10 3
y 10 0
10
3
-10
3
6
iii
3y = 4x
x 3
-
3
-
6
6 0
y 4
-
4
-
8
8 0
Q u e s t i o n : 5
If x = 3 and y = 4 is a solution of the equation 5x – 3y = k, find the value of k.
S o l u t i o n :
Given: 5x – 3y = k
Since x = 3 and y = 4 is a solution of the given equation so, it should satisfy the equation.
5(3)-3(4) = k ? 15 -12 = k ? 3 = k
Q u e s t i o n : 6
If x = 3k + 2 and y = 2k – 1 is a solution of the equation 4x – 3y + 1 = 0, find the value of k.
S o l u t i o n :
Given: 4x – 3y + 1 = 0 .....1
x = 3k + 2 and y = 2k – 1
Putting these values in the equation 1
we get
4(3k +2)-3(2k -1)+1 = 0 ? 12k +8 -6k +3 +1 = 0 ? 6k +12 = 0 ? k +2 = 0 ? k = -2
Q u e s t i o n : 7
The cost of 5 pencils is equal to the cost of 2 ballpoints. Write a linear equation in two variables to represent this statement. (Take
the cost of a pencil to be x and that of a ballpoint to be y).
S o l u t i o n :
Let cost of a pencil to be x and that of a ballpoint to be y.
Cost of 5 pencils = 5x
Cost of 2 ballpoints = 2y
Cost of 5 pencils = cost of 2 ballpoints
? 5x = 2y ? 5x -2y = 0
Page 5
Q u e s t i o n : 1
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
3x + 5y = 7.5
ii
2x -
y
5
+6 = 0
iii
3y – 2x = 6
iv
4x = 5y
v
x
5
-
y
6
= 1
vi
v
2x +
v
3y = 5
S o l u t i o n :
i
3x + 5y = 7.5
This can be expressed in the form ax + by + c = 0 as 3x +5y +(-7. 5) = 0
.
ii
2x -
y
5
+6 = 0
This can be expressed in the form ax + by + c = 0 as 2x + -
1
5
y +6 = 0
iii
3y – 2x = 6
This can be expressed in the form ax + by + c = 0 as 2x +(-3y)+6 = 0
.
iv
4x = 5y
This can be expressed in the form ax + by + c = 0 as 4x -5y +0 = 0
v
x
5
-
y
6
= 1
This can be expressed in the form ax + by + c = 0 as 6x -5y = 30
vi
v
2x +
v
3y = 5
This can be expressed in the form ax + by + c = 0 as
v
2x +
v
3y +(-5) = 0
Q u e s t i o n : 2
Express each of the following equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case.
i
x = 6
ii
3x – y = x – 1
iii
2x + 9 = 0
iv
4y = 7
v
x + y = 4
vi
x
2
-
y
3
=
1
6
+y
( )
S o l u t i o n :
i
x = 6
In the form of ax + by + c = 0 we have x +0y +(-6) = 0
where a = 1, b = 0 and c = -6
.
ii
3x – y = x – 1
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iii
2x + 9 = 0
In the form of ax + by + c = 0 we have 2x +(-1y)+1 = 0
where a = 2, b = -1
and c = 1
iv
4y = 7
In the form of ax + by + c = 0 we have 0x +4y +(-7) = 0
where a = 0, b = 4 and c = -7
v
x + y = 4
In the form of ax + by + c = 0 we have 1x +1y +(-4) = 0
where a = 1, b = 1, c = -4
vi
x
2
-
y
3
=
1
6
+y
In the form of ax + by + c = 0 we have 3x -2y = 1 +6y ? 3x -8y +(-1) = 0
where a = 3, b = -8, c = -1
.
Q u e s t i o n : 3
Check which of the following are the solutions of the equation 5x – 4y = 20.
i
4, 0
ii
0, 5
iii
-2,
5
2
iv
0, – 5
v
2,
-5
2
S o l u t i o n :
The equation given is 5x – 4y = 20.
i
4, 0
Putting the value in the given equation we have
LHS: 5(4)-4(0) = 20RHS: 20LHS = RHS
Thus, 4, 0
is a solution of the given equation.
ii
0, 5
Putting the value in the given equation we have
LHS: 5(0)-4(5) = 0 -20 = -20RHS: 20LHS ? RHS
( )
( )
Thus, 0, 5
is not a solution of the given equation.
iii
-2,
5
2
Putting the value in the given equation we have
LHS: 5(-2)-4
5
2
= -10 -10 = -20RHS: 20LHS ? RHS
Thus, -2,
5
2
is not a solution of the given equation.
iv
0, – 5
Putting the value in the given equation we have
LHS: 5(0)-4(-5) = 0 +20 = 20RHS: 20LHS = RHS
Thus, 0, – 5
is a solution of the given equation.
v
2,
-5
2
Putting the value in the given equation we have
LHS: 5(2)-4
-5
2
= 10 +10 = 20RHS: 20LHS = RHS
Thus, 2,
-5
2
is a solution of the given equation.
Q u e s t i o n : 4
Find five different solutions of each of the following equations:
i
2x – 3y = 6
ii
2x
5
+
3y
10
= 3
iii
3y = 4x
S o l u t i o n :
i
2x – 3y = 6
x 0 3
-
3
9
2
2
y
-
2
0
-
4
1
-2
3
ii
( )
( )
( )
( )
( )
( )
2x
5
+
3y
10
= 3
x 0
15
2
5 10 3
y 10 0
10
3
-10
3
6
iii
3y = 4x
x 3
-
3
-
6
6 0
y 4
-
4
-
8
8 0
Q u e s t i o n : 5
If x = 3 and y = 4 is a solution of the equation 5x – 3y = k, find the value of k.
S o l u t i o n :
Given: 5x – 3y = k
Since x = 3 and y = 4 is a solution of the given equation so, it should satisfy the equation.
5(3)-3(4) = k ? 15 -12 = k ? 3 = k
Q u e s t i o n : 6
If x = 3k + 2 and y = 2k – 1 is a solution of the equation 4x – 3y + 1 = 0, find the value of k.
S o l u t i o n :
Given: 4x – 3y + 1 = 0 .....1
x = 3k + 2 and y = 2k – 1
Putting these values in the equation 1
we get
4(3k +2)-3(2k -1)+1 = 0 ? 12k +8 -6k +3 +1 = 0 ? 6k +12 = 0 ? k +2 = 0 ? k = -2
Q u e s t i o n : 7
The cost of 5 pencils is equal to the cost of 2 ballpoints. Write a linear equation in two variables to represent this statement. (Take
the cost of a pencil to be x and that of a ballpoint to be y).
S o l u t i o n :
Let cost of a pencil to be x and that of a ballpoint to be y.
Cost of 5 pencils = 5x
Cost of 2 ballpoints = 2y
Cost of 5 pencils = cost of 2 ballpoints
? 5x = 2y ? 5x -2y = 0
Q u e s t i o n : 8
Draw the graph of each of the following equations.
i
x = 4
ii
x + 4 = 0
iii
y = 3
iv
y = –3
v
x = –2
vi
x = 5
vii
y + 5 = 0
viii
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