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

Chapter-wise Class 9  Mathematics Linear Equations In Two Variables Worksheets Pdf Download

1. Linear Equations
2. Solution of a Linear Equation
3. Graph of a Linear Equation in Two Variables
4. Equations of Lines Parallel to x-axis and y-axis
• An equation of the form ax + by + c = 0 where a, b and c are real numbers such that a and b are not both zero is called a linear equation in two variables.
•  A pair of values of x and y which satisfy the equation ax + by + c = 0 is called a solution of the equation.
•  Graph: The graph of every linear equation in two variables is a straight line. Every point on the graph of a linear equation in two variables is two variables is a solution of the linear equation. Conversely, every solution of the linear equation is a point on the graph of the linear equation.
•  A linear equation in two variables has infinitely many solutions.
•  The graph of every linear equation in two variables is a straight line.
•  y = 0 is the equation of x-axis and x = 0 is equation of y-axis.
•  The graph of x = a is a straight line parallel to the y-axis.
•  The graph of y = a is a straight line parallel to the x-axis.
•  An equation of the type y = mx represent a line passing through the origin.

Question. If (2, 0) is a solution of the linear equation 2x+ 3y= k, then the value of k is
(a) 4
(b) 6
(c) 5
(d) 2

Answer: A

Question. 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

Answer: A

Question. 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

Answer: B

Question. Any solution of the linear equation 2x + 0y + 9 = 0 in two variables is of the form:
(a) (-9/2,m)
(b) (n,-9/2)
(c) (0,-9/2)
(d) (-9,0)

Answer: A

Question. 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)

Answer: D

Question. Any point on the x-axis is of the form
(a) (x, y)
(b) (0, y)
(c) (x, 0)
(d) (x, x)

Answer: C

Question. The graph of the linear equation 2x+ 3 y= 6 is a line which meets the x-axis at the point
(a) (0, 2)
(b) (2, 0)
(c) (3, 0)
(d) (0, 3)

Answer: C

Question. Any point on the line y = x is of the form
(a) (a, a)
(b) (0, a)
(c) (a, 0)
(d) (a, – a)

Answer: A

Question. The linear equation 2x– 5 y= 7 has
(a) A unique solution
(b) Two solutions
(c) Infinitely many solutions
(d) No solution

Answer: C

Question. The equation of x-axis is of the form
(a) x = 0
(b) y = 0
(c) x + y = 0
(d) x = y

Answer: B

Question. If a linear equation has solutions (–2, 2), (0, 0) and (2, – 2), then it is of the form
(a) y – x = 0
(b) x + y = 0
(c) –2x + y = 0
(d) –x + 2y = 0

Answer: B

Question. 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 distance 6 units from the origin
(c) making an intercept 6 on the x-axis.
(d) making an intercept 6 on both the axes.

Answer: A

Question. The positive solutions of the equation ax + by + c = 0 always lie in the
(a) 1st quadrant
(b) 2nd quadrant
(c) 3rd quadrant
(d) 4th quadrant

Answer: A

Question. x = 5, y= 2 is a solution of the linear equation
(a) x + 2y = 7
(b) 5x + 2y = 7
(c) x + y = 7
(d) 5x + y = 7

Answer: C

Question. The point of the form (a, a) always lies on:
(a) x-axis
(b) y-axis
(c) On the line y= x
(d) On the line x+ y = 0

Answer: C

Question. 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

Answer: C

Question. 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

Answer: D

Question. 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

Answer: B

Question. Every point on the graph of a linear equation in two variables does not represent a solution of the linear equation.
Answer. As every point on the graph of linear equation in two variables represent a solution of the equation, so the given statement is false.

Question. The graph of the linear equation x+ 2y= 7 passes through the point (0, 7).
Answer. Substituting x = 0 and y = 7 in the given equation x + y 2y = 7, we get 0 + 2(7) = 7 ⇒ 14 = 7, which is false.
The point (0, 7) does not satisfy the equation.
Hence, the given statement is false.

Question. Find the solution of the linear equation x+ 2y = 8 which represents a point on
(i) x-axis (ii) y-axis
Answer. We know that the point which lies on x-axis has its ordinate 0.
Putting y = 0 in the equation x + 2y = 8, we get x + 2(0) = 8 ⇒ x = 8
A point which lies on y-axis has its abscissa 0.
Putting x = 0 in the equation x + 2y = 8, we get 0 + 2y = 8 ⇒ y = 4

Question. Write the linear equation such that each point on its graph has an ordinate 3 times its abscissa.
Answer. A linear equation such that each point on it graph has an ordinate 3 times its abscissa is y = 3x. 6. If the point (3, 4) lies on the graph of 3y = ax + 7, then find the value of a.
Answer. The point (3, 4) lies on the graph of 3y = ax + 7.
Substituting x = 3 and y = 4 in the given equation 3y = ax + 7,we get
∴ 3× 4 = a × 3 + 7
⇒ 12 = 3a + 7 ⇒ 3a = 5 ⇒ a = 5/3

Question. How many solution(s) of the equation 2x+ 1 = x– 3 are there on the:
(i) Number line
(ii) Cartesian plane?
Answer. (i) The number of solution(s) of the equation 2x + 1 = x – 3 which are on the number line is one.
2x +1 = x − 3 ⇒ 2x − x = −3 − 1 ⇒ x = −4
∴ x = −4 is the solution of the given equation.
(ii) The number of solution(s) of the equation 2x + 1 = x – 3 which are on the Cartesian plane are infinitely many solutions.

Question. For what value of c, the linear equation 2x+ cy = 8 has equal values of x and y for its solution.
Answer. The value of c for which the linear equation 2x + cy = 8 has equal values of x and y i.e., x = y for its solution is
2x + cy = 8 ⇒ 2x + cx = 8 [∵ y = x]
⇒ cx = 8 − 2x
∵ c = 8 - 2x/x, x ≠ 0

Question. Let y varies directly as x. If y = 12 when x = 4, then write a linear equation. What is the value of y when x = 5?
Answer. y varies of directly as x.
⇒ y ∝ x,
∴ y = kx
Substituting y = 12 when x = 4, we get
12 = k × 4 ⇒ k = 12 ÷ 4 = 3
Hence, the required equation is y = 3x.
The value of y when x = 5 is y = 3 × 5 =15.

Question. Show that the points A (1, 2), B (– 1, – 16) and C (0, – 7) lie on the graph of the linear equation y= 9 x – 7.
Answer. For A (1, 2), we have 2 = 9 (1) – 7 = 9 – 7 = 2
For B (–1, –16), we have –16 = 9(–1) – 7 = – 9 – 7 = – 16
For C (0, –7), we have – 7 = 9 (0) – 7 = 0 – 7 = – 7
We see that the line y = 9x – 7 is satisfied by the points A (1, 2), B (–1, –16) and C (0, –7).
Therefore, A (1, 2), B (–1, –16) and C (0, –7) are solutions of the linear equation y = 9x – 7 and therefore, lie on the graph of the linear equation y = 9x – 7.

Question. The linear equation that converts Fahrenheit (F) to Celsius (^oC) is given by the relation: C = 5F - 160 / 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°C what is the temperature in Fahrenheit and 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?
Answer. C = 5F - 160 / 9
(i) Putting F = 86^o, we get C = 5 (86) - 160 / 9 = 430 - 160 / 9 = 270 / 9 = 30^o
Hence, the temperature in Celsius is 30^o C.
(ii) Putting C = 35^o, we get 35^o 5 (F) - 160 / 9 ⇒315^o = 5F - 160
⇒ 5F = 315 +160 = 475
∴ F = 475 / 5 = 95^o
Hence, the temperature in Fahrenheit is 95 F.
(iii) Putting C = 0^o, we get
0 = 5F - 160 / 9 ⇒ 0 = 5 F - 160
⇒ 5F = 160
∴ F = 160 / 5 = 32^o
Now, putting F = 0^o, we get
C = 5F - 160 / 9 ⇒ C = 5(0) - 160 / 9 = - {160/9}^o
If the temperature is 0^o C, the temperature in Fahrenheit is 32^o and if the temperature is ⁰F, then the temperature in Celsius is - {160/9}^oC
(iv) Putting C = F, in the given relation, we get
F = 5F - 160 / 9 ⇒ 9F = 5F - 160
⇒ 4F = - 160
∴ F = - 160 / 4 = - 40^o
Hence, the numerical value of the temperature which is same in both the scales is – 40.
The linear equation that converts Kelvin (x) to Fahrenheit (y) is given by the relation:
y = 9/5 (x - 273) + 32

Question. If the temperature of a liquid can be measured in Kelvin units as x° K or in Fahrenheit units as y° F, the relation between the two systems of measurement of temperature is given by the linear equation y = 9/5 (x - 273) + 32
(i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313°K.
(ii) If the temperature is 158° F, then find the temperature in Kelvin.
Answer. y = 9/5 (x - 273) + 32
(i) When the temperature of the liquid is x = 313^o K y = 9/5 (313 - 273) + 32 = 9/5 X 40 + 32 = 72^o + 32^o = 104^oF
(ii) When the temperature of the liquid is y = 158^o F 
158 = 9/5 (x - 273) + 32 ⇒ 9/5 (x - 273) = 158 - 32
⇒ x − 273 = 126 × 5/6 = 70
⇒ x − 273 = 70 = 273 + 70 = 343^o K