Linear Equations in Two Variables Class 9 Worksheet with Solutions have been designed as per the latest pattern for CBSE, NCERT and KVS for Grade 9. Students are always suggested to solve printable worksheets for Mathematics Linear Equations in two variables Grade 9 as they can be really helpful to clear their concepts and improve problem solving skills. We at worksheetsbag.com have provided here free PDF worksheets for students in standard 9 so that you can easily take print of these test sheets and use them daily for practice. All worksheets are easy to download and have been designed by teachers of Class 9 for benefit of students and is available for free download.

## Mathematics Linear Equations In Two Variables Worksheets for Class 9

We have provided **chapter-wise worksheets for class 9 Mathematics Linear Equations In Two Variables** which the students can download in Pdf format for free. This is the best collection of Mathematics Linear Equations In Two Variables standard 9th worksheets with important questions and answers for each grade 9th Mathematics Linear Equations In Two Variables chapter so that the students are able to properly practice and gain more marks in Class 9 Mathematics Linear Equations In Two Variables class tests and exams.

### 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 graph of the linear equation y = x passes through the point.

**Answer**

A

**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 given below represents the linear equation x + y= 0.

**Answer.** The given equation is x + y = 0, i.e., y = -x. Any point on the graph of y = – x, will have x and y coordinates of opposite signs. As the points (-1, 1) and (-3, 3) have x and y coordinates of opposite signs, so these points satisfy the given equation and the two points determine a unique line, hence the given statement is true.

**Question.** The graph given below represents the linear equation x = 3 (see Fig. 4.2).

**Answer.** We know that the graph of the equation x = a is a line parallel to the y-axis and to the right of y-axis, if a > 0.

The given statement is true, since the graph is a line parallel to y-axis at a distance of 3 units to the right of it.

**Question.** The point (0, 3) lies on the graph of the linear equation 3x + 4y= 12.** Answer.** Substituting x = 0 and y = 3 in the equation 3 (0) + 4(3) =12 ⇒ 12 =12, which is true.

The point (0, 3) satisfies the equation 3x + 4y = 12.

Hence, the given statement is true.

**Question.** The coordinates of points in the table:

**Represent some of the solution of the equation x – y + 2 = 0.Answer. **The points (0, 2), (1, 3), (2, 4) and (4, 6) satisfy the given equation x – y + 2 = 0. Each of these points is the solution of the equation x – y + 2 = 0. But, the satisfy the given equation as 3 – (–5) + 2 = 0, i.e., 3 + 5 + 2 = 0 or 10 = 0, which is false.

Hence, the given statement is false, since the point (3, -5) does not satisfy the given equation.

**Question.****The graph of every linear equation in two variables need not be a line.****Answer.** As the graph of a linear equation in two variables is always a line, so the given statement is false.

**Question.****Draw the graphs of linear equations y = x and y = – x on the same Cartesian plane. What do you observe?**** Answer.** Any point on the graph of y = x will have x and y coordinates same. The line passes through the points (0, 0), (1, 1) and (-1, -1).

Again, any point on the graph of y = -x will have x and y coordinates of opposite signs. The line passes through the points (1, -1) and (-1, 1). Also, (0, 0) satisfy y = -x.

The graph of linear equation y = x and y = – x on the same Cartesian plane is shown in the figure given below.

We observe that the graph of these equation passes through (0, 0).

**Question.****Determine the point on the graph of the linear equation 2x+ 5y= 19, whose ordinate is 1(1/2) times its abscissa.****Answer.** Let the coordinate of the point be (2, 3).

Now, for x = 2 and y = 3.

2x + 5y = 2(2) + 5(3) = 4 +15 =19

Therefore, the point (2, 3) is a solution of the equation 2x + 5y =19.

Abscissa of the point is 2 and ordinate is 3.

**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.** Draw the graph of the equation represented by a straight line which is parallel to the x-axis and at a distance 3 units below it.**Answer**. The graph of the equation y = – 3 is a line parallel to the axis and at a distance 3 units below it. So, graph of the equation y = – 3 is a line parallel to x-axis and passing through the point (0, -3) as shown in the figure given below:

**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 (^{o}C) 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

^{0}F, then the temperature in Celsius is – {160/9}

^{o}C

(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^{o}F

(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

**Question.** The following observed values of x and y are thought to satisfy a linear equation.

**Write the linear equation.****Draw the graph using the values of x, y given in the above table. At what points the** **graph of the linear equation****(i) cuts the x-axis (ii) cuts the y-axis****Answer.** The linear equation is 2x + 3y = 6. Both the points (6, -2) and (-6, 6) satisfy the given linear equation. Plot the points (-6, 2) and (-6, 6) on a graph paper. Now join these two points and obtain a line. We see that the graph cuts the x-axis at (3, 0) and y-axis at (0, 2).

**Question.****Draw the graph of the linear equation whose solutions are represented by the points having the sum of coordinates as 10 units.****Answer**. A linear equation whose solutions are represented by the points having the sum of coordinates as 10 units is x + y = 10.

When x = 0, y = 10 and when x = 10, y = 0.

Now, plot these two points (0, 10) and (10, 0) on a graph paper and join them to obtain a straight line.

The graph of x = y = 10 is a straight line as shown in the figure given below.

**Question.****Draw the graph of the linear equation 3x+ 4y= 6. At what points, the graph cuts the x-axis and the y-axis.Answer.** The solutions of the linear equation 3x + 4y = 6

Can be expressed in the form of a table as follows by writing the values of y below the corresponding value of x:

Now plot the points (2, 0), (-2, 3) and (0, 1.5) on a graph paper. Now, join the points and obtain a line.

We see that the graph cuts the x-axis at (2, 0) and y-axis at (0, 1.5).

**Question.** The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation in 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) 5 m/sec ^{2},**

**(ii) 6 m/sec**

We have y ∝ x⇒ y = mx

^{2}.**Answer.**Where y denotes the force, x denotes the acceleration and m denotes the constant mass.

Taking m = 6 kg, we get y = 6x

Now, we form a table as follows by writing the value of y below the corresponding value of x.

Plot the points (0, 0), (1, 6) and (2, 12) on a graph paper and join any two points and obtain a line.

## Mathematics Linear Equations In Two Variables Worksheets for Class 9 as per CBSE NCERT pattern

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