linear equations.pdf

8/14/2019 linear equations.pdf

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IMPORTANT TERMS, DEFINITIONS AND RESULTS

l An equation of the form ax+ by+ c= 0, where

a, b, c are real numbers, is called a linear

equation inxandy.For example, 3x+ 2y= 9, 4x

5y= 1 and3

4x 2y= 5 are linear equations in

xand y.

l A linear equation in two variables can be solved

in the same way as a linear equation in one

variable. The pair of values of x and y which

satisfies the given equation is called solution ofthe equation in two variables.

l A linear equation in two variables has infinitely

many solutions.

l In order to draw the graph of a linear equation in

two variables we may follow the following

method :

(i) Expressy in terms of x.

(ii) Choose at least two convenient values ofx

and find the corresponding values of y,

satisfying the given equation.

(iii) Write down these values of x and y in the

form of a table.(iv) Plot the ordered pairs (x, y) from the table

on a graph paper.

(v) Join these points by a straight line and ex-

tend it in both the directions.

This line is the graph of the given equation.

l (i) Equation of a line parallel to they-axis at a

distance a from it is x= a.

(ii) Equation of a line parallel to thex-axis at a

distance b from it is y= b.

1. On putting x = 4, y = 5 in the equation

3x 2y2k= 0, the value of k is :

(a) 5 (b) 2 (c) 11 (d) 11

2. The equation of the line whose graph passes

through the origin, is :

(a) 2x + 3y = 1 (b) 2x + 3y = 0

(c) 2x + 3y = 6 (d) none of these

3. If x = 1, y = 4 is a solution of the equation

mx y = 6, then the value of m is :

(a) 1 (b) 2 (c) 2 (d) 0

4. The general form of a linear equation in two

variables is :

(a) ax+ by+ c= 0, where a, b, care real num-

bers and a, b 0

(b) ax+ b= 0, where a, bare real numbers and

a 0(c) ax2 + bx + c = 0, where a, b, c are real

numbers and a, b 0

(d) none of these

5. If the point 3 1

3,

lies on the graph of the equa-

tion 3y= ax 2, then the value of a is :

(a) 1 (b) 1 (c) 3 (d) 3

6. The solution of 4x y= 5 is :

(a) x= 2, y= 3 (b) x= 3, y = 7

(c) x= 4, y= 11 (d) all the above

SUMMATIVE ASSESSMENT

MULTIPLE CHOICE QUESTIONS [1 Mark]

A. Important Questions

4. Linear Equations in Two Variables

Assignments in Mathematics Class IX (Term 2)


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7. At x = 0, the value of y in the equation

9.35x + 2y = is :

(a)p

2(b) (c) 2 (d) + 2

8. The graph of y= mis a straight line parallel to :

(a) x-axis (b) y-axis

(c) both axes (d) none of these

9. The equation of y-axis is :(a) y = 0 (b) x = 0 (c) y = a (d) x = a

10. The solution of the equation 2x+ 5y= 3 is :

(a) (1, 2) (b) (1, 1) (c) (2, 5) (d) (5, 3)

11. Which of the following ordered pairs is thesolution of the equation 4x 3y= 10 ?

(a) (1, 2) (b) (1, 2)

(c) (1, 2) (d) none of these

12. The value of x for y = 0 in the equation

y = 2x + 1 is :

(a) 2 (b) 1

2(c)

1

2(d) 2

13. If the point (2, 3) lies on the graph of the equa-tion 2y = ax 7, then the value of a is :

(a) 1 (b)1

2(c)

1

2(d)

1

4

14. Any point on the line y= x is of the form :

(a) (a, a) (b) (0, a) (c) (a, 0) (d) (a, a)

15. If we multiply or divide both sides of a linear

equation with a non-zero number, then the

solution of the linear equation :

(a) changes (b) remains the same

(c) changes in case of multiplication only

(d) changes in case of division only

16. The point of the form (a, a) always lies on the

line :

(a) x = a (b) y = a

(c) y = x (d) x + y = 0

17. The linear equation 2x 5y= 7 has :

(a) a unique solution (b) two solutions

(c) infinitely many solutions (d) no solution

18. The equation 2x+ 5y= 7 has a unique solution,

if x, y are :

(a) natural numbers (b) positive real numbers

(c) real numbers (d) rational numbers

19. Any point on the y-axis is of the form :

(a) (x, 0) (b) (x, y) (c) (0, y) (d) (y,y)

20.The graph of which of the following equationsdoes not pass through the origin ?

(a) y x= 2

3(b) y = mx

(c) y = x (d) y = 1

21. If the points (1, a), (b, 15) and (c, 20) lie on

the straight line with equation y = 5x, then the

values of a, band crespectively are :

(a) 4, 5, 3 (b) 5, 3, 4

(c) 3, 4, 5 (d) 1, 4, 5

22. Which of the following lines is not parallel to

y-axis ?

(a) x = 1 (b) x 2 = 0

(c) x + 3 = 0 (d) y+ 1 = 0

23. The linear equation F C=

+9

532 is used to

convert the temperature from Fahrenheit to

Celsius or vice-versa. The temperature numeri-

cally same in both Fahrenheit and Celsius is :

(a) 32 (b) 40

(c) 40 (d) none of these

24. If the points (1, 0) and (2, 1) lie on the graph of

x

a

y

b+ , then the values of a and b are :

(a) a= 1 and b = 1 (b) a= 1 and b= 1

(c) a= 1 and b = 1 (d) a= 0 and b= 1

25. The coordinates of the point where the line

3x 5 = y+ 4 meets the x-axis are :(a) (3, 0) (b) (0, 3)

(c) ( 3, 0) (d) (0, 3)

26. If the points (0, 1) and (1, 0) lie on the graph of

the equation y = mx + c, then the values of m

andc are :

(a) m= 1, c = 1 (b) m= 1, c= 1

(c) m= 1, c = 1 (d) m= 0, c = 1

27. The graph of which of the following is parallel to

x-axis ?

(a) x = a (b) y = a

(c) x = 3 + y (d) y = 0

28. The equation whose graph passes through theorigin is :(a) x + y = 5 (b) x y = 5

(c) x y=1

5(d) none of these

29. The point where the graphs of the equations

x= 3 and y= 3 intersect each other is :

(a) (0, 3) (b) (3, 0)

(c) (3, 3) (d) none of these

30. The value ofyat x= 1 in the equation 5y= 2 is :

(a)5

2

(b)2

5

(c) 10 (d) 0

31. Which of the following is not a linear equation in

two variables ?

(a) p + 4q 7 = 0 (b) u 5v 3 = 0

(c) 2 72

x y( ) (d) 1.2s+ 3t 4 = 032. The force (y) applied on a body is directly

proportional to the acceleration (x) produced in

the body. The linear equation to represent the

given information is :


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(a) xy = k (b) y = kx

(c) x = ky (d) none of these

33. Which of the following is not true about the equa-

tion 2x+ 1 = x 3?

(a) the graph of the line is parallel to y-axis.

(b) x= 4 is the solution of the equation.

(c) point ( 4, 0) lies on the line.

(d) the graph is at right side of the y-axis.

34. The taxi fare in a city is as follows : for the first

kilometre the fare is Rs 10 and for the subsequent

distance it is Rs 7 per km. Taking the distance as

x km and total fare as Rs y, a linear equation

which represents the above information will be :

(a) y = x+ 10 (b) y = 7x + 3

(c) y = 7x (d) none of these

35. How many linear equations in x and y can be

satisfied by x= 1 and y = 2 ?

(a) only one (b) two

(c) infinitely many (d) three

36. The point of the form (a, a) always lies on :(a) x-axis (b) y-axis

(c) the line y= x (d) the line x + y = 0

37. The graph of the linear equation 2x+ 3y= 6 cuts

the y-axis at the point :

(a) (2, 0) (b) (0, 3) (c) (3, 0) (d) (0, 2)

38. The equationx= 7, in two variables, can be writ-

ten as :

(a) 1.x + 1.y = 7 (b) 1.x + 0.y = 7

(c) 0.x + 1.y = 7 (d) 0.x + 0.y = 7

39. The graph of y= 6 is a line :

(a) parallel tox-axis at a distance 6 units from

the origin.

(b) parallel toy-axis at a distnace 6 units fromthe origin.

(c) making an intercept 6 on thex-axis.

(d) making an intercept 6 on both the axes.

40. The positive solutions of the equation ax+ by+

c= 0 always lie in the :

(a) Ist quadrant (b) 2nd quadrant

(c) 3rd quadrant (d) 4th quadrant

41. The graph of the linear equation 2x+ 3y= 6 is a

line which meets the x-axis at the point :

(a) (0, 2) (b) (2, 0) (c) (3, 0) (d) (0, 3)

42. The graph of the linear equation y = x passes

through the point :

(a)3

2

3

2,

(b) 0

3

2,

(c) (1, 1) (d)

1

2

1

2,

B. Questions From CBSE Examination Papers

1. For the equation x 2y= 4, check which of the

following is a solution. [T-II (2011)]

(a) (0, 2) (b) (2, 0) (c) (4, 0) (d) (1, 1)

2. The equation which represents a line which isparallel to x-axis at a distance of 3 cm from the

origin is : [T-II (2011)]

(a)x= 3 and x = 3 (b) y= 3 and y = 3

(c) x= 3 only (d) y= 3 only

3. Which of the following is a solution of the

equation 5x + 2y= 14? [T-II (2011)]

(a)x= 5; y= 1 (b) x = 0; y = 7(c) x= 2; y = 2 (d) x = 1; y = 3

4. The equation x= 4 represents the line which :

(a) is parallel tox-axis [T-II (2011)]

(b) passes through origin

(c) is parallel toy-axis

(c) is perpendicular to y-axis

5. Equation of the line y= 0 represents : [T-II (2011)]

(a)y-axis (b) x-axis

(c) both x-axis and y-axis (d) origin

6. Which of the following lines passes through

(1, 2) ? [T-II (2011)]

(a)x+ y = 3 (b) x y = 1

(c) x = 2 (d) x = 1

7. Which of the following is not a solution of the

equation 2x + y = 7? [T-II (2011)]

(a) (1, 5) (b) (3, 1)

(c) (1, 3) (d) (0, 7)8. Age of x exceeds age of y by 7 years. This

statement can be expressed as linear equation as :

[T-II (2011)](a)x + y+ 7 = 0 (b) x y+ 7 = 0(c) x y 7 = 0 (d) x + y 7 = 0

9. If (2, 0) is a solution of linear equation 2x+ 3y=

k, then the value of k is : [T-II (2011)](a) 4 (b) 6 (c) 5 (d) 2

10. x= 5,y= 2 is a solution of the linear equation :

[T-II (2011)](a)x + 2y = 7 (b) 5x + 2y = 7(c) x + y = 7 (d) 5x + y = 7

11. If point (3, 0) lies on the graph of the equation2x+ 3y= k, then the value of kis :[T-II (2011)](a) 6 (b) 3 (c) 2 (d) 5

12. The equation of x-axis is : [T-II (2011)]

(a)x + y = 0 (b) x y = 0(c) y = 0 (d) x= 0

13. (3, 2) is a point, which belongs to the graph of

the equation : [T-II (2011)]

(a)y = x + 1 (b) 2x = 3y + 1(c) 3x = 2y (d) x= y


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14. The graph of equation of the form ax + by + c

= 0 where a,bare non-zero numbers, represents :

[T-II (2011)]

(a) a triangle (b) a ray(c) a straight line (d) a line segment

15. The condition that the equation ax+ by+ c= 0

represent a linear equation in two variables is :

[T-II (2011)]

(a)a 0, b = 0 (b) b 0, a= 0

(c) a= 0, b= 0 (d) a 0, b 0

16. The linear equation 2y 3 = 0, represented as

ax + by + c= 0, has : [T-II (2011)]

(a) a unique soluiton (b) infinitely many soluitons

(c) two solutions (d) no solution

17. The number of line(s) passing through a point

(3, 4) is / are : [T-II (2011)]

(a) only one (b) two(c) infinite (d) three

18. The graph of the equation ax+ by+ c= 0 may be

of the form : [T-II (2011)]

(a) (b)

(c) (d)

19. The graph of the equation x + a = 0 is a line

parallel to y-axis and to the left of the y-axis if :

[T-II (2011)]

(a) a < 0 (b) a = 0

(c) a > 0 (d) for any real value of a

20. The value of k for which x = 1, y = 1 is a

solution of kx 2y= 0 is : [T-II (2011)]

(a) 12 (b) 2 (c) 5 (d) 8

21. Equation y= 2x+ 3 has : [T-II (2011)]

(a) unique solution

(b) no solution

(c) only two solutions

(d) infinitely many solutions

22. Equation of line parallel to x-axis and 2 units

above the origin is : [T-II (2011)]

(a) x = 2 (b) x= 2 (c) y = 2 (d) y= 2

23. Any point on the line x+ y= 0 is of the form :

[T-II (2011)]

(a) (a, a) (b) (a, a) (c) (0, a) (d) (a, 0)

24. The linear equationx= 5 in two variables can be

written as : [T-II (2011)]

(a) 1.x+ 5 = 0

(b) 0.x+ 1.y+ (5) = 0

(c) 1.x+ 0.y+ (5) = 0

(d) 1.x+ 1.y+ (5) = 0

25. A linear equation in two variables has how many

solutions ? [T-II (2011)]

(a) one (b) two (c) infinite (d) not possible26. The graph of x= 15 is a straight line :

[T-II (2011)]

(a) intersecting both the axes

(b) parallel toy - axis

(c) parallel tox-axis

(d) passing through the origin

27. The point of the form (a, a) always lies on :

[T-II (2011)]

(a) x-axis (b) y-axis

(c) the liney= x (d) the linex+ y= 0

28. Graph of linear equation 4x= 5 in a plane is :

[T-II (2011)](a) parallel tox axis (b) parallel to y-axis(c) lies alongx -axis (d) passes through origin

29. The graph of y= mx is a straight line :

[T-II (2011)]

(a) parallel tox-axis

(b) parallel toy-axis

(c) passing through origin

(d) coincides with x - axis

30. The equation y= 2x+ 5 has : [T-II (2011)]

(a) a unique solution

(b) no solution

(c) infinite solutions(d) only four solutions

31. For what value of k,x= 2 andy= 1 is a solution

of x+ 3y k= 0 ? [T-II (2011)]

(a) 1 (b) 2 (c) 2 (d) 3

32. The point lying on the equation 2x y= 5 is :

[T-II (2011)]

(a) (2, 1) (b) (6, 1) (c) (3, 1) (d) (3, 4)

33. Point P(2, 3) lies on the line represented by the

equation : [T-II (2011)]

(a) x + 2y = 0 (b) 2x + 2y = 0

(c) x + y = 1 (d) 2x + y = 134. The graph of the equation of the form y= mx is

a line which always passes through : [T-II (2011)]

(a) (0, m) (b) (x, 0) (c) (0, y) (d) (0, 0)

35. If a linear equation has solutions (2, 2), (0, 0)

and (2, 2), then it is of the form : [T-II (2011)]

(a) y x= 0 (b) x + y = 0

(c) 2x + y= 0 (d) x+ 2y= 0

36. If the line represented by the equation 3x + y

= 8 passes through the points (2, 2), then the

value of is : [T-II (2011)]

(a) 4 (b) 1 (c) 3 (d) 0


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37. The linear equation 2x+ 5y= 8 has :

[T-II (2011)]

(a) two solutions (b) a unique solution

(c) no solution (d) infinitely many solutions

38. Which of the following represents a line parallel

to x-axis? [T-II (2011)]

(a) x + y = 7 (b)x+ 3 = 0

(c) y+ 2 = 3y 5 (d) 5x+ 3 = 4

39. Any solution of the linear equation 2x+ 0y= 9 in

two variables, is of the form : [T-II (2011)]

(a)9

20,

(b)

9

2, n

, n is a real number

(c) n,9

2

, n is a real number

(d) 0 9

2,

40. Any point on the x-axis is of the form :

[T-II (2011)](a) (0, y) (b) (x, 0) (c) (x, x) (d) (x, y)

41. Linear equation in one variable is : [T-II (2011)]

(a) 2x = y (b)y2 = 3y + 5

(c) 4x y = 5 (d) 3t+ 5 = 9t 7

42. Which of the following is a solution of the

equation 4x + 3y= 16 ? [T-II (2011)]

(a) (2, 3) (b) (1, 4) (c) (2, 4) (d) (1, 3)

43. Graph of the equation 2x+ 3y= 9 cuts y-axis at

the point : [T-II (2011)]

(a)9

20,

(b) (0, 9) (c) (0, 3) (d) (3, 1)

44. Which of the following is not a linear equation ?

[T-II (2011)]

(a) ax + by + c = 0 (b) 0x + 0y + c = 0

(c) 0x + by+ c= 0 (d) ax + 0y + c = 0

SHORT ANSWER TYPE QUESTIONS [2 Marks]


1. Write whether the following statement is true or

false :

The coordiantes of points given in the table

represent some of the solutions of the equation

2x+ 2 = y

x 0 1 2 3 4

y 2 4 6 8 10

2. Express the following equations in the from

ax+ by+ c= 0 and indicate the values of a, bandc.

(i) 5x = y (ii) y = 4x

3. Every point on the graph of a linear equation intwo variables does not represent a solution ofthe linear equation. Is it true ? Justify youranswer.

4. Check whether the graph represents the linearequation x + y= 0 or not.

5. Check whether the point (0, 3) lies on the graph

of the linear equation 3x+ 4y = 12.

6. Find the value of p from the equation

3x + 4y=p, if its one solution isx= 2,y = 1.7. Frame a linear equation in the form ax+ by+

c= 0 by using the given values of a, bandc.

(i) a= 2, b = 3, c = 4

(ii) a = 5, b= 0, c = 78. Find whether the given ordered pair is a

solution of the given linear equation :

(i) 2x 4y= 32; (8, 4)

(ii) 4x 2y= 10; (3, 1)

9. Draw the graph of :(i) x = 4 (ii) y= 3

10. Check whether the graph represents the linear

equation x= 3.

11. Express each of the following equations in the

form y= mx+ c

(i) 3x y= 4 (ii) 2y x = 2

12. Draw the graph of 3x 2y= 0

13. Check whether the graph of the linear equationx+ 2y= 7 passes through the point (0, 7).

14. Draw the graph of the equation y = 2x. From

the graph, find the value of y when x= 2.


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1. Draw the graph of the linear equation in two

variables : 2x + y = 3 [T-II (2011)]

2. If the point (2, 1) lies on the line 5x 2y= 2k ,

find k. Also, find one more solution for the given

equation. [T-II (2011)]

3. The point (3, 4) lies on the graph of the equation

3y = ax+ 7; Find the value of a [T-II (2011)]

4. For what value of k isx= 2, y= 3 a solution of

(k + 1) x (2k+ 3) y 1 = 0? [T-II (2011)]

5. Find the co-ordinates of points where the graph of

equation 4x+ 3y= 12 intersectsx-axis andy-axis.

[T-II (2011)]

6. The auto fares in a city are as follows. For the

first kilometre the fare is Rs 12 and for the

subsequent distance it is Rs 7 per km. Taking the

distance covered as x km and the total fare as

Rs y,write a linear equation. [T-II (2011)]

7. If the point (2k 3, k+ 2) lies on the graph of the

equation 2x+ 3y+ 15 = 0, find the value of k.

[T-II (2011)]

8. Find a value of p for which x= 2, y= 1 is a

solution of the linear equation 5x+ 2py= 2p

[T-II (2011)]

9. Check which of the following is (are) solution (s)

of the equation 3y 2x = 1 [T-II (2011)]

(i) (4, 3) (ii) (2 2 , 3 2 )

10. Find the value of kso that x= 1 and y= 1 is

a solution of the linear equation ikx+ 12ky= 63

[T-II (2011)]

11. Give the equation of one line passing through

(2, 14). How many more such lines are there and

why? [T-II (2011)]

12. If x = 2 and y = 1 is the solution of the linear

equation 2x+ 3y+ k= 0, find the value of k.[T-II (2011)]

13. Express the equation 5x= yin the general form

and indicate the values of a, b and c.[T-II (2011)]

14. How many solution (s) of equation 2x + 1

= x 3 are there : [T-II (2011)]

(a) on number line? (b) in Cartesian plane?

15. Give geometric representation of equation3x+ 12 = 0 in (i) one variable (ii) two variables

[T-II (2011)]

16. Find the point at which the equation 3x 2y= 6

meets the x-axes. [T-II (2011)]

17. Find the coordinates of the points where the line

2x y= 3 meets both the axes. [T-II (2011)]

18. Find four solutions of 2x y= 4. [T-II (2011)]

19. Give two solutions of the equation x+ 3y= 8.

[T-II (2011)]

20. After 5 years, the age of father will be two times

the age of son. Write a linear equation in two

variables to represent this statement. [T-II (2011)]

21. Expressyin terms ofxfrom the equation 3x+ 2y

= 8 and check whether the point (4, 2) lies on

the line. [T-II (2011)]

22. Express 3x= 5y in the form of ax+ by+ c= 0

and hence indicate the values of a, b and c.

[T-II (2011)]

23. If the point (1, 5) lies on the graph of 3x= ay

+ 7, then find the value of a. [T-II (2011)]

24. The cost of a notebook is twice the cost of a pen.

Write a linear equation in two variables to repre-

sent this statement. [T-II (2011)]25. Express y in term of x, given that 2x 5y = 7.

Check whether the point (3, 2) lies on the given

line. [T-II (2011)]

26. Expressyin terms ofxin equation 2x 3y= 12.

Find the points where the line represented by this

equation cuts x-axis and y-axis. [T-II (2011)]

SHORT ANSWER TYPE QUESTIONS [3 Marks]


1. Find the points where the graph of the equation3x + 4y= 12 cuts the x-axis and the y-axis.

2. Draw the graph of the linear equations y = x

andy = x on the same cartesian plane. What

do you observe ?

3. At what point does the graph of the linear

equationx+y= 5 meet a line which is parallel

to the y-axis, at a distnace of 2 units from the

origin in the positive direction of x-axis ?

4. Let y varies directly as x. If y = 12 when

x= 4 then write a linear equation. What is the

value of ywhen x= 5 ?

5. Draw a square whose sides are repreented by

x= 4, x = 4, y = 4, y= 4.

6. Draw a triangle whose sides are represented by

x = 0, y= 0 and x + y = 3.

7. Determine the point on the graph of the linear

equation 2x + 5y = 19 whose ordinate is

11

2times of its abscissa.


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8. Determine the point on the graph of the linearequation 2x + 5y = 20 whose abscissa is

21

2times its ordinate.

9. Draw the graph of the equation represented by

a straight line which is parallel to the

x-axis and at a distnace of 3 units below it.

10. Find three solutions of 5x y + 6 = 0 after

reducing it to y= mx+ cform.

11. Draw the graph of the equation represented by

the striaght line which is parallel to the

x-axis and is 4 units above it.

12. Draw the graph of the linear equation whose

solutions are represented by the points having

the sum of the coordinates as 10 units.

13. Write the linear equation such that each point

on its graph has an ordinate three times its

abscissa.

14. For what value of p, the linear equation

2x+ py= 8 has equal values of xandyfor itssolution?

15. Find the solution of the linear equation

2x+ 5y= 20 which represents a point on

(i) x-axis (ii) y-axis.

1. Draw the graph of the eqation 2x 3y= 5. From

the graph, find the value of ywhen x= 4

[T-II (2011)]

2. Draw the graph of the equation 3x+ 2y= 6. Find

the area of the triangle formed with the line, x-axis and y-axis. [T-II (2011)]

3. Find any three solutions for the equation

15x 2y = 7. [T-II (2011)]

4. Draw the graph x + 2y = 6 and find the points

where the line cuts x-axis and y-axis.

[T-II (2011)]

5. Draw the graph 2x+y= 4 and find the area of the

triangle formed by the line withx-axis andy-axis.

[T-II (2011)]

6. The cost of a table exceeds the cost of the chair

by Rs 150. Write a linear equat ion in twovariables to represent this statement. Also, find

two solutions for the same equation. [T-II (2011)]

7. Draw the graph of the following linear equations :

xy= 7 and 2x=y + 3. At what points does the

graph of each equation cut the x-axis?

[T-II (2011)]

8. If the point (4, 3) lies on the graph of the equation

3x ay= 6, find whether (2, 6) also lies on the

same graph. Find the coordinates of the points

where the graph cuts x-axis and y-axis.

[T-II (2011)]

9. Solve for x : x 1 =3

4 (x+ 1) +

1

2

[T-II (2011)]

10. Draw the graph of equation x y=1 and 2x+ y

= 8 on the same axes. Shade the area bounded by

these lines and x-axis. [T-II (2011)]

11. Draw the graph of the equation 2y x = 7 and

determine from the graph if x = 3, y = 2 is its

solution or not. [T-II (2011)]

12.

Observe the graph and answer the following

questions : [T-II (2011)]

(i) Write the coordinates of point B and C.

(ii) Find one more solution of line passing

through A and B.

(iii) Write equations ofx-axis and y-axis.

13. Draw the graph of the equation y= x+ 1 and

find the point where the graph meets the axes.[T-II (2011)]

14. The linear equation that converts Fahrenheit (F)

to Celsius (C) is given by the relation

C =5 32

9

( )F.

What is the numerical value of the temperature

which is same in both the scales? [T-II (2011)]

15. Draw the graphs of 2x y= 3 and 3x+ 2y = 1

on the same graph paper. Find the point of inter-

section of these graphs. [T-II (2011)]

16. If the number of hours for which a labourer works

is x and y are his wages (in rupees) and

y = 2x 1, draw the graph of the work-wages

equation. From the graph, find the wages of the

labourer if he works for 6 hours. [T-II (2011)]

17. Yamini and Fatima, two students, together

contribute Rs 100 towards the PM Relief Fund to

help the earthquake victims. Write a linear

equation which satisfies this data. Draw the graph

of the same. [T-II (2011)]



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18. Express the lienar equation 2 = 3x in the form

ax + by + c= 0 and indicate the values of a, b

and c. Also give the geometrical representation of

above equation in two variables. [T-II (2011)]

19. Ashish and Deepak contribute to charity. The

contribution of Ashish is2

5 of contribution of

Deepak. Write a linear equation to represent the

above and draw the graph. From the graph, find

the contribution of Ashish, if Deepak contributes

Rs 50. [T-II (2011)]

20. Solve the equation 2y+ 3 = 3y 5 and represent

the solution(s) on :

(i) the number line

(ii) the cartesian plane [T-II (2011)]

21. In countries like USA and Canada, temperature is

measured in Fahrenheit, whereas in countries like

India, it is measured in Celsius. Here is a linear

equation that converts Fahrenheit to Celsius.

[T-II (2011)]

F =9

5

C + 32

(i) If the temperature is 30 C, what is the

temperature in Fahrenheit?

(ii) If the temperature is 95 F, what is the

temperature in Celsius?

(iii) Find the temperature which is numerically the

same in both Fahrenheit and Celsius?

22. Give geometric representation of 2y+ 7 = 0 as an

equation : [T-II (2011)]

(i) in one variable (ii) in two variables

23. Draw a graph of linear equation 3x+ 2y= 12.

[T-II (2011)]

24. Find two different solutions for the linear

equation 3x+ 5y= 15 and check whether (2, 3)

is the solution. [T-II (2011)]

25. Find four solutions of the following linear equa-

tion in two variables. 2(x+ 3) 3 (y 1) = 0

[T-II (2011)]

26. Draw the graph of the equation 3x+ 4y= 12 and

find the coordinates of the points of intersection

of the graph with axes. [T-II (2011)]

27. The taxi fare in a city is charged as per the rates

stated below : [T-II (2011)]

Rate for the first kilometre of joureney is Rs 5

and the rate for the subsequent distance covered

is Rs 4 per km. Taking distance covered as xkm

and total fare as Rsy, write the linear equation in

variablesxandyto express the above statement.

Draw the graph for the linear equation.

[T-II (2011)]

28. Alka and Noori, two students of class IX,

together contributed Rs 500 towards Prime

Ministers Relief Fund to help earthquake victims.

Write a linear equation which satisfies this data

and draw the graph of the same. [T-II (2011)]

29. Find the value of a for which the equation

2x + ay= 5 has (1, 1) as a solution. Find two

more soluiotns for the equation obtained.

[T-II (2011)]

30. Solve the linear equation for x :

2 3

5

3

4

2 3

4

x x x+

+=

+. [T-II (2011)]

31. Find a, ifa

a

a

a

+

= ++

3

2

2

7, a 2, a 7.

[T-II (2011)]

32. Two years later a father will be eight years more

than three times the age of the son. Taking thepresent age of father and son as x and y

respectively,

(a) Write a linear equation for the above and

draw its graph.

(b) From the graph, find the age of father when

sons age is 10 years.

33. Draw the graph of the equation 3x + y = 5 and

write the co-ordinates of the points where the line

intersectsx-axis and y-axis. [T-II (2011)]

34. Draw the graph of the equaiton 2x+ 3y= 6. Write

the points where the line intersects thex-axis and

the y-axis. [T-II (2011)]35. Two pens and three pencils together cost Rs 20.

Represent this statement as a linear equation in

two variables and give two solutions for it. Also,

verify if (4, 2) is a solution of the equation

formed. [T-II (2011)]

36. Draw the graph of the equation represented by a

straight line which is parallel to x-axis and at a

distance 3 units below it. Write its equation also.

[T-II (2011)]

37. Solve the equation 2x+ 1 = x 3 and representthe solution on : [T-II (2011)]

(i) the number line (ii) the cartesian plane

38. Draw the graph of the equation 2(x + 3) 3(1 + y) = 0. Also, find the point where the linemeets x-axis. [T-II (2011)]

39. Give the geometric representations of 2x+ 5 = 0as an equation [T-II (2011)]

(i) in one variable (ii) in two variables.

40. Express the equation y = 2x+ 3 in the standardform and find two solutions. Is (2, 3) its solution?

[T-II (2011)]

41. Sum of the digits of a two digit number is 14. Ifwe add 18 to the original number, the digits inter-

change their places. Write two equations for these

two statements. [T-II (2011)]


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42. The food charges in a hostel are as follows :

For the first day, the charges are Rs 100 and for

the subsequent days it is Rs 50 per day. Taking

the number of days as x and total charges asRs.y, write a linear equation for this information

and draw its graph. [T-II (2011)]

43. Draw the graphs of the equationsx+ y 10 = 0

and x y+ 4 = 0 on the same graph paper.[T-II (2011)]

44. Solve for x :

7 1

4

1

32

1

26

1

3

x

x

x +

= [T-II (2011)]

45. Sum of the digits of a two digit number is 12. If

18 is added to the original number the digits in-

terchange their places. Write two linear equations

representing these situations. [T-II (2011)]

46. Solve the following equation :

x

1

3

1

4

(x 2) = 1 [T-II (2011)]

47. Give geometric representation of x+ 3 = 0 as an

equation (i) in one variable (ii) in two variables.

[T-II (2011)]

48. Solve for x :3 7

5

1

6

2 2

121

x x x

+=

+

[T-II (2011)]

49. Draw the graph of equation 2x+ 3y= 6. From

the graph find the value of xwhen y= 4.

[T-II (2011)]

50. Draw the graph of two lines, whose equations are

3x 2y= 4 and x+ y 3 = 0 on the same graph

paper and find the co-ordinates of the point where

two lines intersect. [T-II (2011)]

51. 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 the triangle formed

by two lines and x-axis. [T-II (2011)]

LONG ANSWER TYPE QUESTIONS [4 Marks]


1. Draw the graph of the linear equation 4x+ y=

6. At what points the graph of the equation cuts

the x-axis and the y-axis ?

2. Draw the graphs of the equations x+y = 6 and

2x + 3y= 16 on the same graph paper. Find the

co-ordinates of the points where the two lines

intersect.

3. Show that the points A (1, 2), B ( 1, 16) andC (0, 7) lie on the graph of linear equation

y = 9x 7.

4. The force exerted to pull a cart is directly

proportional to the acceleration produced in the

body. Express the statement as a linear equation

of two variables and draw the graph of the same

by taking the constant mass equal to 6 kg. Read

from the graph, the force required when the

acceleration produced is (i) 5m/sec2 (ii) 6m/

sec2.

5. The linear equation that converts Fahrenheit (F)

to Celsius (C) is given by the relation

C F

= 5 160

9

(i) If the temperature is 86F, what is thetemperature in Celsius ?

(ii) If the temperature is 35C, what is the

temperature in Fahrenheit ?

(iii) If the temperature is 0F, what is the

temperature in Celsius ?

(iv) What is the numerical value of the

temperature which is same in both the

scales ?

1. The taxi fare in a city is a follows : For the firstkilometer, the fare is Rs, 8 and for the subsequent

distance it is Rs 5/km. Taking the distance

covered as x km and total fare as Rs y, write a

linear equation for this information, and draw its

graph. [T-II (2011)]

2. Solve :4

5

5

6

2

3

1

41

1

6x x+

=

[T-II (2011)]


3. Rs 27 is in the form of 50 paise, 25 paise and20 paise coins. The number of 25 paise coins is

double the number of 20 paise coins but half the

number of 50 paise coins. Find the number of

coins of each type. [T-II (2011)]

4. Solve :y y y

+

=

3

5

4

76

2 1

35

[T-II (2011)]


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5. A number consists of 2 digits. The digit at tens

place is 2 times the digit in units place. The

number formed by reversing the digit is 27 less

than the original number. Find the number.

[T-II (2011)]

6. Solve for x :

x x x+

+=

3

2

3 1

4

2 2

3 2( )

[T-II (2011)]

7. A man leaves half his property to his wife, one

third of the remaining to his son and the rest to his

daughter. If daughters share is Rs 15000, how

much money did the man leave? How much

money did his wife and son each get?

[T-II (2011)]

8. You know that the force applied on a body is

directly proportional to the acceleration produced

in the body. If constant of proportionality is 2,

write an equation to express this situation and plot

the graph of equation. [T-II (2011)]

9. Solve for x :3 2

7

4 1

5

2

3

x x++

+=

( ) (2x + 1)

[T-II (2011)]

10. The total monthly expenditure of a household

consists of a fixed expenditure on house rent as

Rs 500 and the expenditure on rice which is

available at Rs 50 per kg. Write a linear equation

assuming the consumption of rice to be xkg per

month and the total expenditure of the household

per month as Rs y. Draw the graph of the

equaiton. [T-II (2011)]

11. I am three times as old as my son. Five years

later, I shall be two and a half times as old as myson. How old am I and how old is my son?

[T-II (2011)]

12. If the work done by a body on application of a

constant force is directly proportional to the

distance travelled by the body, express this in the

form of an equation in two variables and draw the

graph of the same by taking the constant force as

5 units. Also read from the graph the work done

when the distance travelled by the body is 2 units.

[T-II (2011)]


20

4 5

13

6x x

+ +

= [T-II (2011)]

14. Draw the graph of the linear equation

2x 3y+ 7 = 0 and hence find the coordinates of

the point where the line intersects x-axis.

[T-II (2011)]

15. If (2, 3) and (4, 0) lie on the graph of equation

ax+ by= 1, find the values of aand b. Plot the

graph of equation obtained. [T-II (2011)]

16. Draw the graphs of the equations 2x y= 3 and

3x + 2y= 1 on the same coordinate axes. Also,

find the point of intersection of the two lines from

the graphs. [T-II (2011)]

17. The following values of x and y are thought to

satisfy a linear equation.

x 1 2

y 1 3

Draw the graph, using the values of x and y. At

what points the graph cuts the x-axis and y-axis.

[T-II (2011)]

18. Draw the graph of equation 3x+y= 6. Also, find

the points where the line intersect x-axis and

y-axis. [T-II (2011)]

19. Draw the graphs of each of the equations

x 2y 3 = 0 and 4x+ 3y 1 = 0 on the same

graph. [T-II (2011)]

20. 4 years before, age of a mother was 3 times the

age of her daughter. Write a linear equation to

represent this situation and draw its graph.[T-II (2011)]

21. The parking charges of a car in a parking lot is

Rs. 30 for the first two hours and Rs 10 for

subsequent hours. Taking total parking time to be

xhours and total charges as Rs y, write a linear

equation in two variables to express the above

statement. Draw a graph for the linear equation

and read the charges for five hours. [T-II (2011)]

22. Draw the graph of the linear equation

2x + 3y = 12

(i) Write the co-ordinates of a point where

graph intersectsx-axis.

(ii) From the graph show whether points (3, 2)and (3, 6) are the solution of the equation.

[T-II (2011)]

23. A water tank is getting filled up by water flowing

at the rate of 15 cm3/sec. If the volume of water

filled inyseconds isxcm3, write a linear equation

in two variables to represent this situation. Draw

a graph for the equation formed and hence find

the volume of water filled in 9 seconds.

[T-II (2011)]

24. Draw the graph of the equation 2x+ 3y 6 = 0.

(i) Using graph paper determine whetherx= 3

and y= 0 is a solution

(ii) Find the value ofy, if x= 3 and

(iii) Find the value ofx, ify= 2 from the graph

and verify. [T-II (2011)]

25. Solve for x :x x

= +

+7

46

3 1

22

( )

[T-II (2011)]


6

2 2 7

3

3

2

x x+

+( )=

)

[T-II (2011)]


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27. A and B are friends. A is elder to B by 5 years. Bs

sister C is half the age of B while As father D is

8 years older than twice the age of B. If the

present age of D is 48 years, find the present ages

of A, B and C. [T-II (2011)]


3

4 2

5

25 7

15

x x x+

+=

+

[T-II (2011)]

FORMATIVE ASSESSMENT

Activity

Objective : To draw the graph of a linear equation.

Materials Required : Graph paper, geometry box, etc.

Procedure : Let us draw the graph of the equation

x + 2y = 11

1. Write the given equation x + 2y = 11 as

y =11

2

x

2. Give some suitable values to x and find the

corresponding value of y.

When x= 1, then y =11 1

2

10

25

= =


2

8

24

= =


2

2

21

= =

3. Put the corresponding values of xandyin thetabular form as shown below.

x 1 3 9

y 5 4 1

4. Now plot the points A (1, 5), B (3, 4) and

C(9, 1) on a graph paper.

5. Join AB and extend it on both sides to obtain

the required graph of the equationx+ 2y= 11.

Also, check whether the point C(9, 1) lies on

the line or not.

6. Pick another point on this line say D(5, 3).

Check whether it is a solution of the given

equation or not.

7. Now take any point not lying on the line AB

say E(3, 2). Check whether it is a solution of

the given equation or not.

Observations :

1. After joining AB, if we extend it on both sides,we see that the point C(9, 1) lies on the line

AB. It means every point whose coordinates

satisfy the given equation lies on the line AB.

2. The point D(5, 3) lies on the line AB.

Also, 5 + 2 3 = 5 + 6 = 11. So, (5, 3) is a

solution of the given equation.

Similarly, the point P(7, 2) lies on the line AB.

Also, 7 + 2 2 = 7 + 4 = 11. So, (7, 2) is also

a solution of the given equation.

It implies every point (a, b) on the line AB

gives a solution x = a, y = b of the givenequation.

3. The point E(3, 2) does not lie on the line AB.

Also, 3 + 2 2 = 3 + 4 = 7 11. It implies

any point which does not lie on the line AB is

not a solution of the given equation.

Conclusion : From the above activity, we can conclude

that :

(i) Every point on the line satisfies the equation of

the line.

(ii) Every solution of the equation is a point on the

line.

(iii) Any point which does not lie on the line is not

a soluton of the equation.

Do Yourself : Draw the graphs of each of the following

equations :

1. 2x + y = 5 2. 2x y = 0

3. 4x y = 8 4. 5x 3y = 9

In each case, verify that :

(i) Every point on the line satisfies the equation.

(ii) Every solution of the equation is a point on the

line.

(iii) Any point which does not lie on the line is not

a solution of the equation.

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