RS Aggarwal Class 9 Solutions Linear Equations In Two Variables

 

RS Aggarwal Class 9 Solutions Chapter 8

Q.1: Draw the graph of each of the following equations:

(i) x = 5

(ii) y = -2

(iii) x + 6 = 0

(iv) x + 7 = 0

(v) y = 0

(vi) x = 0

Sol:

(i) The given equation is x = 5

For the given equation, many solutions are possible. Out of them, we take, (5, 1) and (5, -1)

Thus, forming the table:

x 5 5
y 1 -1

On plotting the points, on the graph sheet, the following graph is obtained, in which the line AB is the required graph:

 

https://lh6.googleusercontent.com/PiBe2_TeoWbkFehpGKnj2gaFxCgJq0c3Zfs64t5VfIuYAa5aX2g3RrKhMAJvOl6x-SvYBFnDmSc425SBYKBsWLxraMirRSsaTrmjWkAGQEMWvXOSVTfi0unxApoicIsnB8x_qYSm

 

(ii) The given equation is y = -2

For the given equation, many solutions are possible. Of them, we take, (1, -2) and (3, -2)

Thus, forming the table:

x 1 3
y -2 -2

On plotting the points, on the graph sheet, the following graph is obtained, in which the line CD is the required graph:

https://lh5.googleusercontent.com/OuBQtYHnqifgiK_VJB6wVBGkmktsJHwQVui34AGm2xMF2A-SB0v3gvfLL1iJOPc9TK6Tg3fFiQ4NWI2G1MOuNh6FiGNFwwkmHJaszWggxWH76teztMnpYCt-99xoFyPsZdWFiij_

 

(iii) The given equation is x + 6 = 0

On solving, x = -6

For the given equation, many solutions are possible. Of them, we take, (-6, 1) and (-6, -5)

Thus, forming the table:

x -6 -6
y 1 -5

On plotting the points, on the graph sheet, the following graph is obtained, in which the line EF is the required graph:

 

https://lh6.googleusercontent.com/0Zz4dkQls42u6xOp3hQYHfY0jXLNhU1YwVjiVKyz9eC_VM_1evV8u5dJ1d03OdRpDIuqXbxFCF22o8DK_fb2zcosCfOcRl4VFDHGveTseNYN5Qp_pWjQSqnScHEPpFCTCECSh62r

 

(iv) The given equation is x + 7 = 0

On solving, x = -7

For the given equation, many solutions are possible. Of them, we take, (-7, 6) and (-7, -1)

Thus, forming the table, we get:

x -7 -7
y 6 -1

On plotting the points, on the graph sheet, the following graph is obtained, in which the line GH is the required graph:

 

https://lh3.googleusercontent.com/S_wBS0F5_TKeU7DsOXTENgN61UscNU3ohOHMEP95UPn45qldUt-9-1qojgIDhfd8JSTR3acrMFInEJwCfkkAGV5R93u5Mk7PLP2oDj5o-2esWGjGa41y9UGT1Tqm8Qmwc_WgKJfj

 

(v) The given equation is x = 0, which on plotting using the above method represents the x-axis itself.

(vi) The given equation is y = 0, which on plotting using the above method represents the y-axis itself.

 

Q. 2: Draw the graph of the equation y = 3x. From your graph, find the value of y when x = -2.

Sol:

The given equation is y = 3x.

Assuming the values for any one variable, we can get the value for the other.

Thus, assuming the values of x as 0, 1 and 2 we get,

x = 0 y = 3(0) = 0

x = 1 y = 3(1) = 3

x = 2 y = 3(2) = 6

Forming the table:

x 0 1 2
y 0 3 6

Plotting the points on the graph sheet, we obtain the required graph:

(NOTE: To draw a line, only two points is required. Here, three points are taken for better understanding)

 

https://lh4.googleusercontent.com/CuBIWm33FXuFf8E1qMoY8kY4qAl3-FQpfqspMla230uK1UPRHADrj8tNPTg-fN9h0ob12Yl2xEH-DjPKcmtJJkvcBQwRz2v7HR6MeCz10jJfCcz65UAg5902wu08r8yopmLY3IPS

 

Take a point A on the left of y-axis such that the distance of point A from the y-axis is 2 units.

Draw AB parallel to y-axis cutting the line y = 3x at B. Draw BC parallel to x-axis meeting y-axis at C.

Hence, y = OC = -6

 

Q.3: Draw the graph of the equation x + 2y – 3 = 0. Find your graph, find the value of y when x = 5.

Sol:

The given equation is x + 2y – 3 = 0.

Assuming the values for any one variable, we can get the value for the other.

Thus, assuming the values of y as -2 and 2 we get:

y = -2 x = 3 – 2(-2) = 7

y = 2 x = 3 – 2(2) = -1

Forming the table:

x 7 -1
y -2 2

Plotting the points on the graph sheet, we obtain the required graph:

 

https://lh6.googleusercontent.com/jaqdiTQvbFzQQtXAxkWwoQfvImjkgdyh-UbodVDVbpWrr_WAlZOTF-YBNYBbsYI_k0o8P8kkMoL7LF5cBgkBtVhW2vYLs2nXgJvj8iiio7-OYWKmN2WFPen0AWQ9wOF645oNJL3R

 

Take a point P on x-axis such that OP = 5

Draw PQ parallel to y-axis cutting the line x = 3 – 2y at Q. Through Q, draw PN parallel to x-axis at N. Hence, y = ON = -1

 

Q. 4: Draw the graph of each of the following equations:

(i) y = x

(ii) y = -x

(iii) y + 3x = 0

(iv) 2x + 3y = 0

(v) 3x – 2y = 0

(vi) 2x + y = 0

Sol:

(i) The given equation is y = x

Thus, when x = 1, y = 1 and when x = -1, y = -1

Thus, the table looks as:

x 1 -1
y 1 -1

Plotting the points, the graph is obtained as follows:

 

https://lh4.googleusercontent.com/CLMuCHO65BIDy1fKO5UcuOwZEwyw8ukQ1QrkyDrsTcLP2CQGEDihKDyaJP6roPo5f7wcZJDAg67_khNpntKBoFMYRcurAjh956ra3sMdAis0fzB6YEYppn2ImritW2B7KGwq707Z

 

(ii) The given equation is y = -x.

Thus, when x = 1, y = -1 and when x = -1, y = 1

Thus, the table looks as:

x 1 -1
y -1 1

Plotting the points, the graph is obtained as follows:

 

https://lh3.googleusercontent.com/OkpA0tvDjJLxloJm-Eb68zIcFAJf5GXbSpcVUyBp_57NxoObcH6_ySRii6jvMMOuewreALBzqdYmJK8PBMZeSaCPFxXnQCsVTMwr8ZKk_ukU9PUTX1lbfLwayt3t5nW1pF84NIjy

 

(iii) The given equation is y + 3x = 0.

Solving, y = -3x

Thus, when x = 1, y = -3 and when x = -1, y = 3

Thus, the table looks as:

x 1 -1
y -3 3

Plotting the points, the graph is obtained as follows:

 

https://lh6.googleusercontent.com/gD-cqtL4yCX-GbXGrvob8LvDloHX0cZDhFe8O3rMZR16Yu6FvGs2OSUPmrufUapQoLti3QBOVMXK-m5b6zhi52RN0BdbuFSXwppMUy4awlqPRR6hMxqXnCcMqiKetOZZTqs9Un4C

 

(iv) The given equation is 2x + 3y = 0.

On solving, y=23×x

Thus, when x = 3, y = -2 and when x = -3, y = 2

Thus, the table looks as:

x 3 -3
y -2 2

Plotting the points, the graph is obtained as follows:

 

https://lh3.googleusercontent.com/wsNMCKsJgC_tAQzqxQkEjZCt35vfhJkIs5K0EamlBhnKjJT3Z7Vfweh0e7bvh3SUYPR-jRoXpQnIQnJ_PgAv6U2rH5Jzxy62dHkyoBKMfi7jw2UJyHA3i8nKtpVa9YHWda6CO2uE

 

(v) The given equation is 3x – 2y = 0.

On solving, y=32×x

Thus, when x = 2, y = 3 and when x = -2, y = -3

Thus, the table looks as:

x 2 -2
y 3 -3

Plotting the points, the graph is obtained as follows:

 

https://lh4.googleusercontent.com/Ur6KJ-X8vGzBRdT2BI3FcwGejUbHjstuqeO0T4zbeDg_Zr5KJtJoODqLAm6T9c8tqTuHo9BICqlj0gX3FTPUTZNMD7-7ddN13FcI7OtdJ16JTTi3iNuXd2_9fRb8ZqdYC6pHMV2Q

 

(vi) The given equation is 2x + y = 0.

On solving,  y = -2x.

Thus, when x = 1, y = -2 and when x = -1, y = 2

Thus, the table looks as:

x 1 -1
y -2 2

Plotting the points, the graph is obtained as follows:

 

https://lh5.googleusercontent.com/zNPekGEcQfl1eDZffk9O-WWZBWVM_mJ-T1vkDHs-hs_g2krg0SbmmF_8CdD9nYDCSbWtfV_mpptHAp0YDoQtX6q1W6Q2QuWIbTQZyXFrePeVQru9wptmXj7qRY-UmwMogmqkQ2TM

 

Q.5: Draw the graph of the equation 2x – 3y = 5. From the graph, find (i) the value of y when x = 4, and (ii) the value of x when y = 3

Sol:

The given equation is 2x – 3y = 5.

On solving, y=2x53

Thus, when x = 4, y = 1 and when x = -2, y = -3

Thus, the table looks as:

x 4 -2
y 1 -3

Plotting the points, the graph is obtained as follows:

 

https://lh4.googleusercontent.com/BRGrmwOz5x0yRKC28-doPcgNc-rNIlG_LTzhEV-Jqw-2D5LXMPSyN7eZj3u6OdAvsxgyM6u1G-IxipnTO66Pe3df2rscEEE4UbZcnTKn1mQLLZblibgrlQorTo9ihDwLkqo_2Rtv

 

Q.6: Draw the graph of the equation 2x + y = 6. Find the coordinates of the point, where the graph cuts the x-axis.

Sol:

The given equation is 2x + y = 6.

On solving, y = 6 – 2x

Thus, when x = 1, y = 4 and when x = -1, y = 8

Thus, the table looks as:

x 1 -1
y 4 8

Plotting the points, the graph is obtained as follows:

 

https://lh6.googleusercontent.com/PJSjLxmNpREkKewtYMAVMcJ2oKyl5ll-KYJ9i4XdAI3r3cH3w55xj_MAU6CUxHHcg6UBau6qH7iIDCMl9aqM51QCSz9g1BYa8xRqYwGLIoPXG2_ryLOtgLRRbJRN7Ur3sTGb9t_4

 

We find that the line cuts the x-axis at a point P which ia at a distance of 3 units to the right of y-axis.

Therefore, the coordinates of P are (3, 0).

 

Q.7: Draw the graph of the equation 3x + 2y = 6. Find the coordinates of the point, where the graph cuts the y-axis.

Sol:

The given equation is 3x + 2y = 6.

On solving, y=63x2

Thus, when x = 0, y = 0 and when x = -2, y = 6

Thus, the table looks as,

x 0 -2
y 0 6

Plotting the points, the graph is obtained as follows:

We find that the given line cuts the y-axis at a point Z which is 3 units above the x-axis.

Therefore, the coordinates of Z are (0,3).

 

https://lh3.googleusercontent.com/id5MpNZgO69tGGJtlRwtv8xOzzwZUTGfq0w4p9RXt4KLBJOP8oQWn653I8BDm9lYlMXWY-IumQHa_ZxwK3Ix_GksLuTElTk5VFPT0HB8DOZVkxXTZD6EjGIcnmcKykF42GoDyXK2

 

CCE Questions:

Question 1:

(b) the y-axis.

X = 0 is the equation of the y-axis.

Therefore, the graph of x = 0 is YOY’

Question 2:

(a) the x-axis.

Y = 0 is the equation of x-axis.

Therefore, the graph of y = 0 is X’OX.

Question 3:

(b) parallel to the y-axis and passing through (-3, 0)

Given equation: x + 3 = 0.

Or, x = -3.

The equation of the line is parallel to the y-axis and passes through (-3, 0).

Question 4:

(b) parallel to the x-axis and passing through (0, 4)

Given equation: y – 4 = 0.

Or, y = -4.

The equation of the line parallel to the x-axis and passes through (0, 4).

Question 5:

(c) the line y = x

Given, a point of the form (a, a), where a 0.

When a = 1, the point is (1, 1)

When a = 2, the point is (2, 2)…. And so on.

When these points i.e., (1, 1), (2, 2)…. And so on are plotted, and the line obtained is extended on both the sides, the equation of the line y = x is obtained.

Question 6:

(d) the line x + y = 0.

Given, a point of the form (a, -a), where a 0.

When a = 1, the point is (1, -1)

When a = 2, the point is (2, -2)…. And so on.

When these points i.e., (1, 1), (2, 2)…. And so on are plotted, and the line obtained is extended on both the sides, the equation of the line x + y = 0 is obtained.

Question 7:

(c) infinitely many solutions.

Given linear equation: 3x – 5y = 15

Or, x=5y+153

When y = 0, x = 5

When y = 3, x = 10

When y = -3, x = 0

Thus, we have the following table:

x 5 10 0
y 0 3 -3

Plotting the points (5, 0), (10, 3) and (0, -3) and joining them and extending both the ends, we get infinite points that satisfy the given equation.

Hence, the linear equation has infinitely many solutions.

 

https://lh5.googleusercontent.com/pBuEsSfTzfZBMvoeUJuOJlQcdkM7Bxu6XCsgNcm7oWoIhv8xPWLdsMCTBil-K_-HUVg2YMoZKYXWqhg8nK8xcSi7S1dBm5__9US_BZL02W1fPYASWQaOl4uOHw13xDcHFHyYR7OX

 

Question 8:

(c) (0, 3)

Given linear equation: 3x + 2y = 6

Or, y=63x2

Putting, x = 0, we get y = 3

Thus, the graph cuts the y-axis at (0, 3)

Question 9:

(d) (3, 0)

Given equation: 4x + 3y = 12

Or, x=123y4

When, y =0, x = 3

Thus, the graph cuts the x-axis at (3, 0)


Practise This Question

Which of the following is a viral disease?