CBSE Class 9 Mathematics Linear Equations In Two Variables Worksheet Set C

Question. In which of the following equations will the value of y decrease as the value of x increases?
A. y = x-1
B.  y = 2x-15
C. y = 5-2x
D. y = x/3

Answer : C

Question. One man takes one day to dig a 4 m long trench. How long would it take 2 men working at the same rate to dig a 16 m long trench?
A. 1 day
B.  2 days
C. 4 days
D. 8 days

Answer : B

Question. Akbar, Ali and Arman are 3 brothers. The ratio of Akbar's age to Arman's age is 1 : 2 and the ratio of Akbar's age to Ali's age is 2 : 5. If the eldest boy is 10 years old, how old is the youngest one?
A. 1 year
B. 2 years
C. 4 years
D. 5 years

Answer : C

Question. Suparna has saved up Rs. 538 for a new pair of jeans. The price of the pair she wants is Rs. 750. Suparna's mother tells her that she can earn Rs. 15 per hour by helping her with the housework. Which of the following will help Suparna figure out how many hours (x) of housework she needs to do before she has enough to buy the jeans?
A. 538 + 15x = 750
B. 15x = 750
C. 750 - 538x =15
D. (538 + 15)x = 750

Answer : A

Question. This is a sale notice that Mr. Rai has put up in his shop to attract customers. However, not really intending to sell his goods at a low price, he has marked each item at 10% higher than the actual marked price before announcing the sale. At what price would he be selling an item whose original marked price was P?

A. P
B. 10%of P
C. 90%of P
D. 99% of P

Answer : D

Question. I started two clocks at the same time. One runs slow and loses 1 minute every hour. The other one is fast and gains 3 minutes every hour. How long will it take for the faster clock to be exactly one hour ahead of the slower clock?
A. 1 day
B. 18 hours
C. 15 hours
D. 12 hours

Answer : C

Question. In the weighing scales shown below, the same shapes represent the same weights. According to the balanced scales P and Q, what weights should be put on the pan on the right to balance scale R?

Answer : B

Question. P, Q and R are three friends. Q's height is 5/6 times the height of P. R's height is 1/5 times that of Q. Which of these statements is true?
A. The ratio of P's height to that of Q is 6 : 5.
B. P is shorter than Q
C. Q is shorter than R
D. P and R are of the same height

Answer : A

Question. Points P, Q and R are co-planar. In which of the following cases will they NECESSARILY be collinear?
A. When PQ = PR
B. When PQ + PR > QR
C. When PQ + QR = PR
D. When PR < PQ + QR

Answer : B

Question. For a quadrilateral PQRS inscribed in a circle, with  PQ || RS, which of the following is NOT necessarily true?

A. ∠Q + ∠ P = 180⁰
B. Q +S = 180⁰
C. Q +  R = 180⁰
D. S +  P = 180⁰

Answer : A

Question. In the following figure  PR || AC, QP || AB and RQ || BCIf the perimeter of triangleABC is 24 cm, the perimeter of triangle PQR will be

A. 6 cm
B. 8 cm
C. 12 cm
D. 16 cm

Answer : C

Question. If the length of the longer line is 60 cm, the length of the shorter one is
l .........................................
m........................................
A. 25 cm
B. 22.5 cm
C. 20 cm
D. 18 cm

Answer : B

Question. The clocks below show the date and time in two different places in the world at the same, who stays in Mumbai, wants to chat with a friend who stays in Los Angeles on the internet. Everyday, George is on the internet from 9 a.m. to 8 p.m. (Mumbai time) and his friend is on the internet from 6 a.m. to 7 p.m. (Los Angeles time). According to the local time in Mumbai, what would be a suitable time for them to chat?

A. 9.00 a.m. to 9.30 a.m
B. 6.00 p.m. to 7.00 p.m.
C. 7.30 p.m. to 8.00 p.m.
D. 9.00 a.m. to 7.00 p.m.

Answer : C

Question. The difference between two temperature readings in an experiment was 90Which of these could be the temperature readings?
A. 1 and -6^0.
B. -5⁰ and 4^0.
C. -1^0.and 9^0.
D. -2^0.and 11^0.

Answer : B
 

ASSERTION & REASONING QUESTIONS

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R).
Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : A linear equation 3x + 5y = 2 has a unique solution.
Reason : A linear equation in two variables has infinitely many solutions.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : We know that a linear equation in two variables has infinitely many solutions.
So, Reason is correct.
Hence, Assertion is not correct
Correct option is (d) Assertion (A) is false but reason (R) is true.

Question. Assertion : If x = 2, y = 1 is a solution of the equation 2x + 3y = k, then the value of k is 7.
Reason : The solution of the line will satisfy the equation of the line.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : We know that the solution of the line will satisfy the equation of the line.
So, Reason is correct.
Since x = 2, y =1 is a solution of the given linear equation, we have
2 x 2 + 3 x 1 – k = 0
So, Assertion is also correct
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .

Question. Assertion : If x = 2k – 1 and y = k is a solution of the equation 3x – 5y – 7 = 0, then the value of k is 10
Reason : A linear equation in two variables has infinitely many solutions.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : We know that a linear equation in two variables has infinitely many solutions. So, Reason is correct.
Since x = 2k - 1 and y = k is solution of the given linear equation, we have
3 x (2k – 1) – 5k – 7 = 0
So, Assertion is also correct
But reason (R) is not the correct explanation of assertion (A) .
Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) .

Question. Assertion : There are infinite number of lines which passes through (3, 2) .
Reason : A linear equation in two variables has infinitely many solutions.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : We know that a linear equation in two variables has infinitely many solutions. So, Reason is correct.
Through a point infinite lines can be drawn.
Through (3, 2) infinite number of lines can be drawn.
Hence, Assertion is also correct.
But reason (R) is not the correct explanation of assertion (A) .
Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) .

Question. Assertion: x = 3 and y = 2 is a solution of the linear equation 2x + 3y = 12.
Reason: x = 4 and y = 2 is a solution of the linear equation x + 3y = 10.
Answer : For Assertion: The given linear equation is 2x + 3y = 12
Substituting x = 3 and y = 2, we get
LHS = 2 x 3 + 3 x 2 = 6 + 6 = 12 = RHS
So, Assertion is correct.
For Reason: The given linear equation is x + 3y = 10
Substituting x = 4 and y = 2, we get
LHS = 4 + 3 x 2 = 4 + 6 = 10 = RHS
So, Reason is also correct.
But reason (R) is not the correct explanation of assertion (A) .
Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) .

Question. Assertion : The point (3, 0) lies on the graph of the linear equation 4x + 3y = 12.
Reason : (3, 0) satisfies the equation 4x + 3y = 12.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : For Assertion: The given linear equation is 4x + 3y = 12
Substituting x = 3 and y = 0, we get
LHS = 4 x 3 + 3 x 0 = 12 + 0 = 12 = RHS
Since (3, 0) satisfies the equation 4x + 3y = 12, therefore point (3, 0) lies on the graph of the linear equation 4x + 3y = 12.
So, Reason and Assertion are both correct.
Here, Reason is the correct explanation of Assertion.
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .

Question. Assertion : The graph of the linear equation 2x – y = 1 passes through the point (2, 3) .
Reason : Every point lying on graph is not a solution of 2x – y = 1.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : For Assertion: The given linear equation is 2x – y = 1
Substituting x = 2 and y = 3, we get
LHS = 2 x 2 – 3 = 4 – 3 = 1 = RHS
Since (3, 0) satisfies the equation 4x + 3y = 12, therefore graph of the linear equation 2x – y = 1 passes through the point (2, 3) .
So, Assertion is correct.
But Reason is not correct as every point lying on graph is a solution of 2x – y = 1.
Correct option is (c) Assertion (A) is true but reason (R) is false.

Question. Assertion: x = 2 is a line parallel to the y-axis.
Reason: The equation of a line parallel to the y-axis is x = a.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : We know that equation of a line parallel to the y-axis is x = a.
So, Reason (R) is true.
Also, x = 2 is a line parallel to the y-axis.
So, Assertion (A) is true.
Thus, Reason (R) and Assertion (A) are true and Reason (R) is a correct explanation of Assertion (A) .
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .

Question. Assertion: x + y = 3 is the equation of a line passing through the origin.
Reason: y = 2x is the equation of a line passing through the origin.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : For Assertion: The given linear equation is x + y = 3
Since x = 0 and y = 0 is not satisfying x + y = 3, therefore it is not passing through the origin.
So, Assertion is not correct.
Since x = 0 and y = 0 is not satisfying y = 2x, therefore it is passing through the origin.
So, Reason is correct.
Correct option is (d) Assertion (A) is false but reason (R) is true.

Question. Assertion: y = 3x represents a line passing through the origin.
Reason: Any line parallel to the x-axis is y = a.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A) .
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : Since x = 0 and y = 0 is not satisfying y = 3x, therefore it is passing through the origin.
So, Assertion (A) is true.
Also, we know that equation of a line parallel to the x-axis is y = a.
So, Reason (R) is also true.
But Reason is not the correct explanation of Assertion.
Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) .

Question. Assertion : The point (2, 2) is the solution of x + y = 4.
Reason : Every point which satisfy the linear equation is a solution of the equation.
Answer : We know every point which satisfy the linear equation is a solution of the equation.
So, Reason (R) is true.
Substituting x = 2 and y = 2, we get
LHS = 2 + 2 = 4 = RHS
Since (3, 0) satisfies the equation 4x + 3y = 12, therefore the point (2, 2) is the solution of x + y = 4
So, Assertion (A) is also true.
Here, Reason is the correct explanation of Assertion.
Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) .

1. (a) Give the equations of two lines passing through (3, - 2) . How many more such lines are there, and why?
(b) Solve the equation 2x + 5 = x + 2/5 , and represent the solution(s) on (i) the number line(ii) the Cartesian plane.
(c) Give the geometric representations of 2y – 9 = 0 as an equation (i) in one variable (ii) in two variables.

2. Draw geometric representations of 2x + 3y = 5. Check whether the points (- 3, 4) and (7, - 3) are solutions of the given equation.

3. Draw the graph of the following equations. Read two more solutions from the graph in each case. Also, find the coordinates of the points where the line intersects the two axes:-
(a) x – 3y = 6
(b) 2x – 5y = -10
(c) 3x + 4y = - 12.

4. Find 4 solutions of each of the following equations:-
(a) x + 2y = - 5
(b) 2(x – 1) + 3 = 5(1 – y)
(c) 2x + 3y = 6
(d) 4x + 20 = 5y.

5. Find a value of ‘a’ such that :-
a. x = 3, y = - 2, is a solution of the equation 5x – 2ay = 5. Now, find one more solution of this equation.
b. x = - 1, y = 4 is a solution of the equation 2ax – 3y = 8. Now, find one more solution of this equation.
c. x = 1, y = 1 is a solution of the equation 5x – 2ay = 3a. Now, find one more solution of this equation.
d. x = 3, y = 4 is a solution of the equation 5ax + 12ay = 63.Now, find one more solution of this equation.

6. Draw the graphs of the following equations on the same pair of axes:-
(1) 2y + 5 = 0 (2) x = 4 (3) 3x + 12 = 0
(4) y – 5 = 0 (5) x – y = 0 (6) 2x + y = 0.

7. The taxi fare in a city is as follows: For the first two kilometres, the fare is Rs.25 and for the subsequent distance it is Rs.10 per km. Taking the distance covered as x km and the total fare as Rs.y, write the linear equation representing this situation and draw its graph. Also, from the graph, determine the fare that a person will have to pay for covering a distance of 10km.

1. Find two solutions of the linear equation 5x – 4y = - 8

2. Draw the graph of the linear equation 2x + 3y = 12. At what points the graph of the equation cuts the x – axis and the y axis

3. Draw the graphs of the equations x + y = 6 and 2x + 3y = 16 on the same graph paper. Find the coordinates of the points where the two lines intersect

4. Draw the graph of the following equation 2( x + 1) = 3 ( y – 1) – 4 and check whether the point (3, -1) lies on the line

5. Draw the graph of y = - 5 and y = 5 on the same graph. Are the lines parallel? Find the point of intersection of two lines

6. The taxi fare in a city is such that Rs 50 is fixed amount and Rs 16 per km is charged. Taking the distance covered as x km and total fare as Rs y, write a linear equation in x and y

7. If present age of son and father are expressed by x and y respectively and after ten years father will be twice as old as his son. Write the relation between x and y

8. If the cost of 5 tables exceeds the cost of eight chairs by Rs. 150. Write the linear equation in two variables to represent the statement. Also find the cost of 1 table if the cost of one chair is RS. 240

9. Give the geometric representation of 2x + 1 = x – 4 as an equation in (a)one variable (b) two variable

10. Give the equation of two lines passing throw (2, 14). How many more such lines are there and why

11. If (2, 5) is a solution of the equation 2x + 3y = m, find the value of m (m = 19)

12. For what value of k does the point (k, -3) lies on the line 3x – y = 6 (k = 1)

13. Write 13x -12y = 25 as y= mx + c. Hence find m and c. Verify if x = 1, y = 1 is a solution (m= 13/12, c = - 25/12)

14. If (2, 3) and (4, 0) lie on the graph of the equation ax + by = 1. Find the value of a and
b. Plot the graph of the equation obtained ( a =1/4, b = 1/6)

15. Express y in terms of x, given that x/5 + 2y = 3. Check whether (-5, 2) is asolution of the given equation

16. Write each of the following as an equation in two variables (in standard form):
(a) x = - 5
(b) y = 2
(c) 2x = 3
(d) 5y = 2

16. Frame a linear equation in the form ax + by + c = 0 by using the given values of a, b and c : a = -2 , b = 3, c = 4

17. Solve for x :
a) (3x+2)/7 + 4 (x+1)/5 = 2( 2x + 1)/3        (x = 4)
b) 8x + 21/4 = 3x+ 7                                  (7/20)

19. Graph of linear equation 4x = 5 in a plane is parallel to ……….axis

20. When the linear equation 2x = 3/8(y – 1) is written in the standard form ax + by + c = 0
Then a, b, c are …….. , ……… and ………

21. The geometric representation of 2y + 5 = 0 in two variables is a straight line parallel to …….. axis

22. Coefficient of y in the equation: 3(2x -1/3y) = 0 is equal to a) 3 b) 1 c)-3 d)-1

23. Alinear equation in two variable has
a) infinitely manysolution
b) unique solution
c) no solution
d) none of these

24. Which of the following pair is a solution of the equation 2x – 3y = 7
a) (5,-1)
b) (1, 5)
c) (0, 2)
d) none of these

25. The equation of a line passing through the origin is of the form
a) y = k x
b) x + y = k
c) x – y = k
d) none of these

26. Any point on y axis is of the form
a) (x, 0)
b) (0, y)
c) (y, 0)
d) none of these

27. The graph of y = mx is a straight line:
a) parallel to x axis
b) parallel to y axis
c) passing throw origin
d) coincides with x – axis

28. For the equation 5x + 8y = 50, if y = 10, then the value of x is
a) 6
b) – 6
c) 12
d) – 12

29. The equation x = 7, in two variables can be written as:
a) 1x +1 y = 7
b) 1x + 0y = 7
c) 0x + 1y = 7
d) 0x + 0y =7

30. Equation of line parallel to x – axis and 2 – units above the originis:
a) x = 2
b) x = -2
c) y = 2
d) y = -2

31. Which of the following is not a form of linear equation in two variables?
a) ax +by + c = 0
b) ax + 0y + b = 0
c) 0x + ay + b = 0
d) 0x + 0y + 5 = 0

Q1: Find the value of K if x=2 and y=1 is a solution of the equation (k-2)x + 4y = 10 and have the graph of the equation. Write the co-ordinate of the point where the graph cuts the y-axis.

Q2: Express y in terms of x in the equation 2x-3y = 12. Find the points where the line represented by this equation cuts x-axis and y- axis.

Q3: Draw the graph of two lines whose equations are 3x-2y -6 =0 and x+2y-6=0 on the same graph paper. Find the area of triangle formed by the two lines and x-axis.

Q4: Find the solutions of the form x=a, y=0 and x=0, y=b for the following equations:

2x+5y = 10 and 2x+3y = 6. Is there any common solution?

Q5: If the points A(3,5) and B(1,4) lie on the graph of the line ax+by = 7, find the values of a and b.

Q6: Draw the graphs of 2x+ y=6 and 2x-y +2 =0. Shade the region bounded by these lines and x-axis. Find the area of the shaded region.

Q7: Draw the graphs of x- y=1 and 2x+y =8. Shade the region bounded by these lines and y-axis. Shade the area bounded by these two lines and y-axis. Also, determine this area.

Q8: Ravish tells his daughter Aarushi, “Seven years ago, I was seven times as old as you will be”. If present ages of Aarushi and Ravish are x and y respectively, represent this situation algebraically as well as graphically.

Q9: If the points A(3,5) and B(1,4) lie on the graph of the line ax+ by = 7, find the values of a and b.

Q10: Draw the graphs of each of the following linear equations in Cartesian plane

(i) x-2=0 (ii) 2x + 4 = 3x + 1