Download Linear Equations in Two Variables

Linear Equations in Two Variables
IMPORTANT TERMS, DEFINITIONS AND RESULTS
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An equation of the form ax + by + c = 0, where
a, b, c are real numbers, is called a linear
equation in x and y. For example, 3x + 2y = 9, 4x
– 5y = 1 and
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3
x – 2y = 5 are linear equations in
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(v) Join these points by a straight line and extend it in both the directions.
This line is the graph of the given equation.
(i) Equation of a line parallel to the y-axis at a
distance a from it is x = a.
x and y.
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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 of
the equation in two variables.
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A linear equation in two variables has infinitely
many solutions.
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In order to draw the graph of a linear equation in
two variables we may follow the following
method :
(i) Express y in terms of x.
(ii) Choose at least two convenient values of x
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.
(ii) Equation of a line parallel to the x-axis at a
distance b from it is y = b.
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SUMMATIVE ASSESSMENT
MULTIPLE CHOICE QUESTIONS
[1 Mark]
A. Important Questions
(b) ax + b = 0, where a, b are real numbers and
a ≠ 0
(c) ax2 + bx + c = 0, where a, b, c are real
numbers and a, b ≠ 0
1. On putting x = 4, y = –5 in the equation
3x – 2y –2k = 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
(d) none of these
⎛ 1⎞
5. If the point ⎜⎝ 3, ⎟⎠ lies on the graph of the equa3
tion 3y = ax – 2, then the value of a is :
(a) – 1
4. The general form of a linear equation in two
variables is :
(a) ax + by + c = 0, where a, b, c are real numbers and a, b ≠ 0
(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
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ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
7. At x = 0, the value of y in the equation
9.35x + 2y = π is :
p
(a)
(b) π
(c) π – 2 (d) π + 2
2
8. The graph of y = m is 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)
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
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23. The linear equation F = ⎛⎜ ⎞⎟ C + 32 is used to
⎝ 5⎠
convert the temperature from Fahrenheit to
Celsius or vice-versa. The temperature numerically 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
11. Which of the following ordered pairs is the
solution 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 :
1
1
(a) 2
(b) –
(c)
(d) – 2
2
2
13. If the point (2, – 3) lies on the graph of the equation 2y = ax – 7, then the value of a is :
1
1
1
(a) 1
(b)
(c) –
(d)
2
2
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 equations
does not pass through the origin ?
2
(a) y = x
(b) y = mx
3
(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, b and c respectively are :
(a) 4, 5, – 3
(b) – 5, 3, – 4
(c) 3, 4, 5
(d) – 1, 4, 5
x y
+ , then the values of a and b are :
a b
(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)
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26. If the points (0, 1) and (1, 0) lie on the graph of
the equation y = mx + c, then the values of m
and c 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 the
origin is :
(a) x + y = 5
(b) x – y = 5
1
(c) x = y
(d) none of these
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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
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30. The value of y at x = 1 in the equation 5y = 2 is :
5
2
(b)
(c) 10
(d) 0
2
5
31. Which of the following is not a linear equation in
two variables ?
(a) p + 4q – 7 = 0
(b) πu – 5v – 3 = 0
(a)
(c)
(
2x − 7 y
)
2
(d) 1.2s + 3t – 4 = 0
32. 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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ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
(a) xy = k
(b) y = kx
(c) x = ky
(d) none of these
33. Which of the following is not true about the equation 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.
38. The equation x = 7, in two variables, can be written 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 to x-axis at a distance 6 units from
the origin.
(b) parallel to y-axis at a distnace 6 units from
the origin.
(c) making an intercept 6 on the x-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 :
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)
⎛ 3 −3 ⎞
(a) ⎜⎝ , ⎟⎠
2 2
(b)
⎛ 3⎞
⎜⎝ 0, ⎟⎠
2
(c) (1, 1)
(d)
⎛ −1 1 ⎞
⎜⎝ , ⎟⎠
2 2
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B. Questions From CBSE Examination Papers
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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)
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2. The equation which represents a line which is
parallel 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
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 equation
2x + 3y = k, then the value of k is : [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
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 to x-axis
[T-II (2011)]
(b) passes through origin
(c) is parallel to y-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
(c) x = 2
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)
(b) x – y = 1
(d) x = 1
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ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
14. The graph of equation of the form ax + by + c
= 0 where a, b are 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
(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 possible
26. The graph of x = 15 is a straight line :
[T-II (2011)]
(a) intersecting both the axes
(b) parallel to y - axis
(c) parallel to x-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 line y = x
(d) the line x + y = 0
28. Graph of linear equation 4x = 5 in a plane is :
[T-II (2011)]
(a) parallel to x axis (b) parallel to y-axis
(c) lies along x -axis (d) passes through origin
29. The graph of y = mx is a straight line :
[T-II (2011)]
(a) parallel to x-axis
(b) parallel to y-axis
(c) passing through origin
(d) coincides with x - axis
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)
(c)
(b)
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(d)
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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
30. The equation y = 2x + 5 has :
(a) a unique solution
(b) no solution
(c) infinite solutions
(d) only four solutions
[T-II (2011)]
31. For what value of k, x = 2 and y = – 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 = 1
34. 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)
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)
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
24. The linear equation x = 5 in two variables can be
written as :
[T-II (2011)]
(a) 1.x + 5 = 0
(b) 0.x + 1.y + (–5) = 0
36. If the line represented by the equation 3x + αy
= 8 passes through the points (2, 2), then the
value of α is :
[T-II (2011)]
(a) 4
(b) 1
(c) 3
(d) 0
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ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
37. The linear equation 2x + 5y = 8 has :
[T-II (2011)]
(a) two solution s (b) a unique solution
(c) no solution
(d) infinitely many solutions
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)
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)]
⎛9
(a) ⎜⎝ ,
2
⎞
0⎟
⎠
43. Graph of the equation 2x + 3y = 9 cuts y-axis at
the point :
[T-II (2011)]
⎛9 ⎞
(b) ⎜⎝ , n⎟⎠ , n is a real number
2
⎛9 ⎞
(a) ⎜⎝ , 0⎟⎠ (b) (0, 9)
2
⎛ 9⎞
(c) ⎜⎝ n, ⎟⎠ , n is a real number
2
(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
⎛ 9⎞
(d) ⎜⎝ 0, ⎟⎠
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SHORT ANSWER TYPE QUESTIONS
[2 Marks]
A. Important Questions
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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
y
0
2
1
4
2
6
3
8
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6. Find the value of p from the equation
3x + 4y = p, if its one solution is x = 2, y = 1.
7. Frame a linear equation in the form ax + by +
c = 0 by using the given values of a, b and c.
(i) a = – 2, b = 3, c = 4
(ii) a = 5, b = 0, c = 7
8. 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.
2. Express the following equations in the from
ax + by + c = 0 and indicate the values of a, b
and c.
(i) 5x = – y
(ii) y = – 4x
3. Every point on the graph of a linear equation in
two variables does not represent a solution of
the linear equation. Is it true ? Justify your
answer.
4. Check whether the graph represents the linear
equation x + y = 0 or not.
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 equation
x + 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.
5. Check whether the point (0, 3) lies on the graph
of the linear equation 3x + 4y = 12.
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ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
B. Questions From CBSE Examination Papers
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 is x = 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 intersects x-axis and y-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)]
10.
11.
12.
13.
15.
16.
17.
18.
19.
20.
21. Express y in terms of x from the equation 3x + 2y
= 8 and check whether the point (4, –2) lies on
the line.
[T-II (2011)]
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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)]
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(i) (4, 3)
(ii) (2 2 , 3 2 )
Find the value of k so that x = –1 and y = –1 is
a solution of the linear equation ikx + 12ky = 63
[T-II (2011)]
Give the equation of one line passing through
(2, 14). How many more such lines are there and
why?
[T-II (2011)]
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)]
Express the equation 5x = –y in the general form
L
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14.
and indicate the values of a, b and c.
[T-II (2011)]
How many solution (s) of equation 2x + 1
= x – 3 are there :
[T-II (2011)]
(a) on number line?
(b) in Cartesian plane?
Give geometric representation of equation
3x + 12 = 0 in (i) one variable (ii) two variables
[T-II (2011)]
Find the point at which the equation 3x – 2y = 6
meets the x-axes.
[T-II (2011)]
Find the coordinates of the points where the line
2x – y = 3 meets both the axes. [T-II (2011)]
Find four solutions of 2x – y = 4. [T-II (2011)]
Give two solutions of the equation x + 3y = 8.
[T-II (2011)]
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)]
A
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 represent 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. Express y in terms of x in 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]
A. Important Questions
x = 4 then write a linear equation. What is the
value of y when 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
1. Find the points where the graph of the equation
3x + 4y = 12 cuts the x-axis and the y-axis.
2. Draw the graph of the linear equations y = x
and y = – x on the same cartesian plane. What
do you observe ?
3. At what point does the graph of the linear
equation x + 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
1
1
times of its abscissa.
2
6
ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
8. Determine the point on the graph of the linear
equation 2x + 5y = 20 whose abscissa is
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 x and y for its
solution?
15. Find the solution of the linear equation
2x + 5y = 20 which represents a point on
(i) x-axis
(ii) y-axis.
1
times its ordinate.
2
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 + c form.
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.
2
B. Questions From CBSE Examination Papers
12.
1. Draw the graph of the eqation 2x – 3y = 5. From
the graph, find the value of y when 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, xaxis and y-axis.
[T-II (2011)]
3. Find any three solutions for the equation
15x – 2y = 7.
[T-II (2011)]
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IA
4. Draw the graph x + 2y = 6 and find the points
where the line cuts x-axis and y-axis.
[T-II (2011)]
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O
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L
5. Draw the graph 2x + y = 4 and find the area of the
triangle formed by the line with x-axis and y-axis.
[T-II (2011)]
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 of x-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
I
N
A
6. The cost of a table exceeds the cost of the chair
by Rs 150. Write a linear equation in two
variables to represent this statement. Also, find
two solutions for the same equation. [T-II (2011)]
7. Draw the graph of the following linear equations :
x – y = 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)]
C=
5(F − 32)
.
9
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 intersection 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)]
3
1
(x + 1) +
4
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)]
9. Solve for x : x – 1 =
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)]
7
ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
28. Alka and Noori, two students of class IX,
together contributed Rs 500 towards Prime
Minister’s 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 :
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
2
of contribution of
5
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)]
contribution of Ashish is
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)]
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⎛ 9⎞
F = ⎜⎝ ⎟⎠ C + 32
5
(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?
Give geometric representation of 2y + 7 = 0 as an
equation :
[T-II (2011)]
(i) in one variable (ii) in two variables
Draw a graph of linear equation 3x + 2y = 12.
[T-II (2011)]
Find two different solutions for the linear
equation 3x + 5y = 15 and check whether (2, 3)
is the solution.
[T-II (2011)]
Find four solutions of the following linear equation in two variables. 2(x + 3) – 3 (y – 1) = 0
[T-II (2011)]
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)]
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 x km
and total fare as Rs y, write the linear equation in
variables x and y to express the above statement.
Draw the graph for the linear equation.
[T-II (2011)]
R
O
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U
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22.
23.
24.
25.
26.
27.
2x − 3 x + 3 2x + 3
+
=
. [T-II (2011)]
5
4
4
a+3 a+2
31. Find a, if
, a ≠ 2, a ≠ 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 the
present 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
son’s 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
intersects x-axis and y-axis.
[T-II (2011)]
34. Draw the graph of the equaiton 2x + 3y = 6. Write
the points where the line intersects the x-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 represent
the 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 line
meets x-axis.
[T-II (2011)]
39. Give the geometric representations of 2x + 5 = 0
as an equation
[T-II (2011)]
(i) in one variable (ii) in two variables.
40. Express the equation y = 2x + 3 in the standard
form and find two solutions. Is (2, 3) its solution?
[T-II (2011)]
41. Sum of the digits of a two digit number is 14. If
we add 18 to the original number, the digits interchange their places. Write two equations for these
two statements.
[T-II (2011)]
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8
ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
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 as
Rs. y, write a linear equation for this information
and draw its graph.
[T-II (2011)]
43. Draw the graphs of the equations x + y – 10 = 0
and x – y + 4 = 0 on the same graph paper.
[T-II (2011)]
44. Solve for x :
7x − 1 1 ⎛
1
x − 1⎞
− ⎜ 2x +
⎟⎠ = 6
⎝
4
3
2
3
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 :
49. Draw the graph of equation 2x + 3y = 6. From
the graph find the value of x when y = 4.
[T-II (2011)]
[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)]
45. Sum of the digits of a two digit number is 12. If
18 is added to the original number the digits interchange their places. Write two linear equations
representing these situations.
[T-II (2011)]
46. Solve the following equation :
⎛ x − 1⎞ 1
⎜⎝ 3 ⎟⎠ − 4 (x – 2) = 1
3x − 7 x + 1 2 x + 2
−
=
−1
5
6
12
[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)]
[T-II (2011)]
LONG ANSWER TYPE QUESTIONS
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A. Important Questions
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) and
C (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
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acceleration produced is (i) 5m/sec2 (ii) 6m/
sec2.
R
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TU
A
5. The linear equation that converts Fahrenheit (F)
to Celsius (C) is given by the relation
5F − 160
C=
9
(i) If the temperature is 86°F, what is the
temperature in Celsius ?
(ii) If the temperature is 35°C, what is the
temperature in Fahrenheit ?
(iii) If the temperature is 0°F, what is the
temperature in Celsius ?
(iv) What is the numerical value of the
temperature which is same in both the
scales ?
B. Questions From CBSE Examination Papers
1. The taxi fare in a city is a follows : For the first
kilometer, 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⎞ 2
⎜ x + ⎟⎠ −
5⎝
6
3
3. Rs 27 is in the form of 50 paise, 25 paise and
20 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)]
1⎞
1
⎛
⎜⎝ x − ⎟⎠ = 1
4
6
[T-II (2011)]
4. Solve :
y −3 y −4
⎡ 2 y − 1⎤
+
=6−⎢
⎥
5
7
⎣ 35 ⎦
[T-II (2011)]
9
ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
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.
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 + 3 3 x + 1 2( x − 2)
−
=
−2
2
4
3
x
1
2
y
1
3
[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 daughter’s 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)]
3
x
+
2
4
(
x
+
1
)
2
9. Solve for x :
+
= (2x + 1)
3
7
5
[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 x kg 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 my
son. 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)]
18.
19.
20.
21.
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)]
Draw the graph of equation 3x + y = 6. Also, find
the points where the line intersect x-axis and
y-axis.
[T-II (2011)]
Draw the graphs of each of the equations
x – 2y – 3 = 0 and 4x + 3y – 1 = 0 on the same
graph.
[T-II (2011)]
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)]
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
x hours 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)]
Draw the graph of the linear equation
2x + 3y = 12
(i) Write the co-ordinates of a point where
graph intersects x-axis.
(ii) From the graph show whether points (3, 2)
and (–3, 6) are the solution of the equation.
[T-II (2011)]
A water tank is getting filled up by water flowing
at the rate of 15 cm3/sec. If the volume of water
filled in y seconds is x cm3, 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)]
Draw the graph of the equation 2x + 3y – 6 = 0.
(i) Using graph paper determine whether x = 3
and y = 0 is a solution
(ii) Find the value of y, if x = – 3 and
(iii) Find the value of x, if y = –2 from the graph
and verify.
[T-II (2011)]
x−7
3( x + 1)
−6=
+2
Solve for x :
4
2
[T-II (2011)]
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R
O
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22.
I
N
A
23.
24.
2 x − 10 4 x + 5
+
= 6 [T-II (2011)]
20
13
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 a and 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,
13. Solve for x :
25.
26. Solve for x :
4 x + 5 2 ( 2 x + 7) ) 3
=
−
2
6
3
[T-II (2011)]
10
ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261
27. A and B are friends. A is elder to B by 5 years. B’s
sister C is half the age of B while A’s 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)]
28. Solve for x :
3 x − 5 4 x + 2 25 x + 7
+
=
3
5
15
[T-II (2011)]
FORMATIVE ASSESSMENT
6. Pick another point on this line say D(5, 3).
Check whether it is a solution of the given
equation or not.
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
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 :
11 − x
2
2. Give some suitable values to x and find the
corresponding value of y.
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.
y=
When x = 1, then y =
11 − 1 10
=
=5
2
2
When x = 3, then y =
11 − 3 8
= =4
2
2
11 − 9 2
= =1
When x = 9, then y =
2
2
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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.
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OR
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3. Put the corresponding values of x and y in the
tabular form as shown below.
x
1
A
9
y
5
4
1
3
4. Now plot the points A (1, 5), B (3, 4) and
C(9, 1) on a graph paper.
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 given
equation.
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.
5. Join AB and extend it on both sides to obtain
the required graph of the equation x + 2y = 11.
Also, check whether the point C(9, 1) lies on
the line or not.
11
ANIL TUTORIALS,D-156,SECTOR-5,DEVENDRA NAGAR,RAIPUR,CHHATTISGARH,HELPLINE : 97525-09261