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QUESTION: 1

A system of two linear equations in two variables has infinitely many solutions, if their graphs

Solution:

A system of two linear equations in two variables has infinitely many solutions, if their graphs coincide with each other

QUESTION: 2

The area of the triangle formed by the lines 2x + y = 6, 2x – y + 2 = 0 and the x – axis is

Solution:

Here are the two solutions of each of the given equations. 2x+y = 6

2x+y=0

The area bounded by the given lines and x−axis has been shaded in the graph.

∴

QUESTION: 3

The system of equations x – 4y = 8, 3x – 12y = 24

Solution:

Given

Therefore, the system has infinite many solution.

QUESTION: 4

The system of linear equations a_{1}x+b_{1}y+c_{1 }= 0 and a_{2}x+b_{2}y+c_{2} = 0 has a unique solution if

Solution:

_{1}x+b_{1}y+c_{1 }= 0 and a_{2}x+b_{2}y+c_{2} = 0 has a unique solution if

QUESTION: 5

The sum of the numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. The fraction is

Solution:

QUESTION: 6

A system of two linear equations in two variables is consistent, if their graphs

Solution:

A system of two linear equations in two variables is consistent, if their graphs intersect only at a point, because it has a unitque solution.

QUESTION: 7

The vertices of the triangle formed by the lines x – y + 1 = 0, 3x + 2y – 12 = 0 and the x – axis are

Solution:

Here are the two solutions of each of the given equations.x−y+1=03x+2y−12=0

The triangle formed by given two linear equations is shaded in the graph. The triangle formed by given lines is ΔAEF Therefore, the vertices of triangle AEF are A ( –1, 0), E (2, 3) and F (4, 0).

QUESTION: 8

The pair of equations 5x – 15y = 8 and 3x - 9y = 24/5 has

Solution:

Given: a_{1} = 5,a_{2} = 3,b_{1} = −15,b_{2} = −9,c_{1} = 8 and

Therefore, the the pair of given linear equations has infinitely many solutions.

QUESTION: 9

If x = α and y = β is the solution of the equations x – y = 2 and x + y = 4, then

Solution:

Given: x−y = 2…..(i)

And x+y = 4 ………(ii)

Adding eq. (i) and (ii) for elimination of y, we get

2x = 6 ⇒ x = 3

Putting the values of x in eq. (i), we get

3−y = 2 ⇒ y = 1

∴x = α = 3 and y = β = 1

QUESTION: 10

Half the perimeter of a rectangular garden, whose length is 4m more than its width is 36m. The area of the garden is

Solution:

Let the length of rectangular garden be x meters and breadth be y meters. According to question,

x = 4+y ⇒ x−y = 4 ……….(i)

Also half of the perimeter = 36 ⇒ x+y = 36 ……….(ii)

Adding eq. (i) and (ii), we get 2x = 40 ⇒ x = 20 meters

Subtracting eq. (i) from (ii), we get 2y = 32 ⇒ y = 16 meters

∴ Area = Length x Breadth = 20 x 16 = 320 sq. meters

QUESTION: 11

A system of two linear equations in two variables is dependentconsistent, if their graphs

Solution:

A system of two linear equations in two variables is dependentconsistent, if their graphs coincide with each other.

QUESTION: 12

One equation of a pair of dependent linear equations is – 5x + 7y = 2, then the second equation can be

Solution:

If equation of a pair of dependent linear equations, then

For satisfying the condition of dependent linear equations, the values of a_{2},b_{2} and c_{2} should be the multiples of the values of a_{1},b_{1} and c_{1}.

∴ the values would be and c_{1} = 2×(−2)=−4

∴ the second equation can be 10x−14y = −4

QUESTION: 13

The pair of equations x = 2 and y = – 3 has

Solution:

Here a unique solution of each variable of a pair of linear equations is given, therefore, it has one solution of a system of linear quations.

QUESTION: 14

The solution of px + qy = p – q and qx – py = p + q is

Solution:

QUESTION: 15

The difference between two numbers is 26 and one number is three times the other. The numbers are

Solution:

Let the two numbers be x and y.

According to question, x−y = 26 and x = 3y

Putting the value of x in x−y = 26 we get 3y−y = 26 ⇒ y = 13 And x = 3 × 13 = 39

QUESTION: 16

The lines representing the pair of equations 5x – 4y + 8 = 0 and 7x + 6y – 9 = 0

Solution:

QUESTION: 17

One equation of a pair of inconsistent linear equations is 2x – 3y = 4, then the second equation can be

Solution:

If equation of a pair of inconsistent linear equations, then

For satisfying the condition of dependent linear equations, the values of a_{2},b_{2} and c_{2} should be the multiples of the values of a_{1},b_{1} and c_{1}.

∴ the values would be

∴ the second equation can be 4x−6y = 8

QUESTION: 18

The value of ‘k’ so that the system of equations 3x – 4y – 7 = 0 and 6x – ky – 5 = 0 have a unique solution is

Solution:

QUESTION: 19

Solution:

QUESTION: 20

The sum of two numbers is 35 and their difference is 13. The numbers are

Solution:

Let the numbers be x and .y.

According to question, x+y = 35 ………(i) and x−y = 13 ………..(ii)

Adding eq. (i) and (ii), we get 2x = 48 ⇒ x = 24

Putting the value of x in eq. (i), we get 12+y = 35 ⇒ y = 13

Therefore, the numbers are 24 and 13.

QUESTION: 21

What will be the solution of these equations ax+by = a-b, bx-ay = a+b

Solution:

QUESTION: 22

The area of the triangle formed by x + 3y = 6, 2x – 3y = 12 and the y – axis is

Solution:

Here are the two solutions of each of the given equations. x+3y=62x−3y=12

QUESTION: 23

The value of ‘k’ so that the system of linear equations kx – y – 2 = 0 and 6x – 2y – 3 = 0 have no solution is

Solution:

QUESTION: 24

The pair of linear equations ax + by = c and px + qy = r has a unique solution then

Solution:

Given: a_{1} = a,a_{2} = p,b_{1} = b,b_{2} = q,c_{1} = c and c_{2 }= r

Since the pair of given linear equations has a unique

QUESTION: 25

The value of ‘a’ so that the point (3, a) lies on the line represented by 2x – 3y = 5 is

Solution:

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