Worksheet on Linear Equations in Two Variables Class 10 have been designed as per the latest pattern for CBSE, NCERT and KVS for Grade 10. Students are always suggested to solve printable worksheets for Mathematics Linear Equations Grade 10 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 10 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 10 for benefit of students and is available for free download.

## Mathematics Linear Equations Worksheets for Class 10

We have provided **chapter-wise worksheets for class 10 Mathematics Linear Equations** which the students can download in Pdf format for free. This is the best collection of Mathematics Linear Equations standard 10th worksheets with important questions and answers for each grade 10th Mathematics Linear Equations chapter so that the students are able to properly practice and gain more marks in Class 10 Mathematics Linear Equations class tests and exams.

### Chapter-wise Class 10 Mathematics Linear Equations Worksheets Pdf Download

**Question. In the equations 3x + 2y = 13xy and 4x -5y = 2xy , the value of xand y satisfy that the equations are: **

(A) (2,3)

(B) (3,2)

(C) (1/2,1/3)

(D) (1/3,1/2)

**Answer**

C

**Question. A father is 7 times as old as his son. Two year ago, the father was 13 times as old as his son. Father’s present age is: **

(A) 24 years

(B) 28 years

(C) 30 years

(D) 32 years

**Answer**

B

**Question. If x + y + 7and3x -2y=11. Then the value of x will be: **

(A) 5

(B) 6

(C) 7

(D) 8

**Answer**

A

**Question. If 3y -2x = 4and4y – px = 2perpendicular to each other the value of ‘p’ will be: **

(A) 3/2

(B) 8/3

(C) 6

(D) -6

**Answer**

D

**Question. A boat whose speed is 18km/hr in still water takes 1 hr more to go 24km upstream than to return downstream to the same spot. Find the speed of the stream. **

(A) 8 km/hr

(B) 6 km/hr

(C) 10 km/hr

(D) 5.5 km/hr

**Answer**

B

**Question. It is given that there is no solution to the system x + 2y = 3,ax +by = 4 . Which one of the following is true? **

(A) a has a unique value

(B) b has a unique value

(C) a can have more than one value

(D) a has exactly two different values

**Answer**

C

**Question. In a two digit number, the number of ten’s place is double of the number of unit’s place. If we exchange the numbers mutually then the number decrease b 18, then the number is:- **

(A) 24

(B) 36

(C) 39

(D) 42

**Answer**

D

**Question. The system of equation- x + 2y = 6,3x +6y=18 **

(A) Is inconsistent

(B) Has an infinite number of solution

(C) Has a unique solution

(D) None of these

**Answer**

B

**Question. A man can row three quarters of a km against the stream in 11(1/4) minutes and return in 7(1/2) minutes. The speed of man in still water is: **

(A) 2 km/h

(B) 3 km/h

(C) 4 km/h

(D) 5 km/h

**Answer**

D

**Solved Problems on Class 10 Linear Equations**

**Question. What are the coordinates of points where two lines representing the given equations meet y-axis?**

**Answer**

(0, –2), (0, 4)

**Question. The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.**

**Answer**

4/7

**Question**. Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. Find the present ages of father and son.

**Answer**

Father’s age = 42 years, Son’s age = 10 years

**Question**. Determine graphically, the vertices of the triangle formed by the lines y = x, 3y = x, x + y = 8.

**Answer**

(0, 0), (4, 4) (6, 2)

**Question. If a1/a2≠b1/b2≠c1/c2 , then what does the system of linear equations, represent graphically?**

**Answer**

Two coincident lines

**Question**. The cost of 4 pens and 4 pencil boxes is ` 100. Three times the cost of a pen is ` 15 more than the cost of a pencil box. Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box.

**Answer**

10, ` 15

**Question**. **The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.**

**Answer**

36

**Question. Define consistent system of linear equations.**

## An**swer**

A pair of linear equations which has either unique or infinitely many solutions.

**Question. What is the solution of given pair of equations? Read from graph.**

**Answer**

Unique solution: (3, 1)

**Question**. The sum of a two digit number and the number obtained by reversing the order of its digits is 165. If the digits differ by 3, find the number.

**Answer**

69 or 96

**Question** A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of the stream.

**Answer**

2.5 km/h

**Question**. **Points A and B are 70 km apart on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other, they meet in one hour. Find the speed of the two cars.**

**Answer**

Speed from point A = 40 km/ h, from point B = 30 km/h

**Question**. **Yash scored 35 marks in a test, getting 2 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?**

**Answer**

25

**Question**. **Draw the graphs of the equations y = -1, y = 3 and 4x – y = 5. Also, find the area of the quadrilateral formed by the lines and the y-axis.**

**Answer**

6 square units

**Question. What does a linear equation in two variables represent geometrically?**

## An**swer**

A straight line.

**Question. What is the area of triangle formed by given lines and x-axis?**

**Answer**

1 sq. unit

**Question**. A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

**Answer**

100 km/h, 80 km/h

**Question**. Solve graphically the system of linear equations:**(ii) 4x-3y+4=0****4x-3y+20= 0****Find the area bounded by these lines and x-axis.**

**Answer**

x = 2, y = 4, 12 sq. units

**Question**. Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if she travels 2 km by rickshaw, and the remaining distance by bus. On the other hand, if she travels 4 km by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and of the bus.

**Answer**

10 km/h, 40 km/h

**Question. When is a system of linear equations called inconsistent?**

**Answer**

When it has no solution.

**Question**.**A part of monthly hostel charges in a college are fixed and the remaining depend on the number of days one has taken food in the mess. When a student A takes food for 15 days, he has to pay 1200 as hostel charges whereas a student B, who takes food for 24 days, pays 1560 as hostel charges. Find the fixed charge and the cost of food per day.**

**Answer**

₹ 600, ₹ 40

**Question**. Solve the following system of linear equations graphically and shade the region between the two lines and x-axis.**(i) 3x+2y-4=0 (ii) 3x+2y-11=0****2x-3y+10= 0 2x-3y-7= 0**

**Answer**

(i) x = 2, y = -1 (ii) x =1, y = 4

**Question. Do the equations x +2y -7 = 0 and 2x + 4 y +5= 0 represent a pair of parallel lines?**

**Answer**

Yes

**Question. 2 women and 5 men can together finish a piece of embroidery in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone, and that taken by 1 man alone to finish the embroidery.**

**Answer**

One man in 36 days, One woman in 18 days

**Question. Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum respectively. She received 1860 as annual interest. However, had she interchanged the amount of investment in the two schemes, she would have received 20 more as annual interest. How much money did she invest in each scheme?**

**Answer**

Scheme A ₹ 12000, Scheme B ₹ 10,000

**Question. What is the area of triangle formed by given lines and y-axis?**

**Answer**

9 sq. units

**Question. The car hire charges in a city comprise of a fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is 89 and for a journey of 20 km, the charge paid is person have to**

**Answer**

₹ 215

**Question. Is it true to say that the pair of equations x +2y -3 = 0 and 3x +6 y -9 = 0 are dependent?**

## A**nswer**

Yes

**Question. A two digit number is 4 times the sum of its digits and twice the product of the digits. Find the number.**

**Answer**

36

**Question. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of two cars?**

**Answer**

60 km/h, 40 km/h

**Question. If lines corresponding to two given linear equations are coincident, what can you say about the solution of the system of given equations?**

**Answer**

It has infinitely many solutions.

**Question. What are the coordinates of points where two lines representing the given equations meet x-axis?**

**Answer**

(2, 0), (4, 0)

**VERY SHORT ANSWER TYPE QUESTIONS**

**Question. For what value of k, the equations 2x – ky + 3 = 0 and 3x + 2y – 1 = 0 has no solution?****Solution.** -4/3

**Question. Is x = 2 and y = – 1 a solution of the pair of linear equations x + 2y = 0 and 3x + 4y = 20?****Solution.** No

**Question. What are the coordinates of the point of intersection of two lines 3x + 2y = 0 and 2x – y = 0?****Solution.** x = 0, y = 0

**Question. How many solutions do two linear equations in two variables have, if their graph are parallel?****Solution.** No solution

**Question. At what point does the line 4x + 5y = 20 intersects x-axis?****Solution.** (5, 0)

**Question. How many solutions will the following pair of linear equations have?****7x – 4y + 11 = 0****2x – 9y + 15 = 0****Solution.** unique

**Question. The graph of y = – 3 is a straight line parallel to which axis?****Solution.** x-axis

**Question. Draw the graph of the equations****4x – y= 4****4x + y = 12****Determine the vertices of the triangle formed by the lines representing these equations and the x-axis. Shade the triangular region so formed****Solution.** (2, 4)(1,0 )(3,0 )

**Question. Find the value of ‘a’ so that the point (2,9) lies on the line represented by ax-3y=5****Solution.** a=16

**Question. Determine the value of ‘a’ if the system of linear equations 3x+2y-4=0and ax–y–3=0 will represent intersecting lines.****Solution.** a ≠ 3/2

**Question. Write any one equation of the line which is parallel to √2x–√3y=5****Solution.** 5√2x – 5√3 y = 5√5

**Question. Check whether given pair of lines is consistent or not 5x–1=2y,y= -1/2 + 5/2 x .****Solution.** Consistent

**Question. Find the fraction which becomes to 2/3 when the numerator is increased by 2 and equal to 4/7 when the denominator is increased by 4.****Solution.** 28/45

**Question. Solve Graphically****x – y = –1****3x + 2y = 12.****Calculate the area bounded by these lines and the x- axis.****Solution.** x = 2, y = 3 and area = 7.5 unit^{2}

**Question. Solve for u& v****4u–v=14uv****3u + 2v = 16uv where u≠0, v≠ 0****Solution.** u = ½ , v = ¼

**Question. Ritu can row downstream 20 km in 2 hours, and upstream 4km in 2hours. Find her speed of rowing in still water and the speed of the current.****Solution. **Speed of the rowing in still water = 6km/hr

Speed of the current=4km/hr.

**Question. Write the condition so that a _{1}x + b_{1}y = c_{1} and a_{2}x + b_{2}y = c_{2} have unique solution.**

**Solution.**a

_{1}/a

_{2}≠ b

_{1}/b

_{2}

**Question. 5 pencils and 7 pens together cost Rs.50 whereas 7 pencils and 5 pens together cost Rs.46.Find the cost of one pencil and that of one pen.****Solution.** Cost of one pencil=Rs.3

Cost of one pen=Rs.5

**Question. Form a pair of linear equations for the following problem :****Fourteen students of class X took part in a quiz and the number of boys is 2 more than the number of girls.****Solution.** Let x and y denote the number of boys and girls respectively. Then, x = y + 2 and x + y = 14.

**Question. What is the minimum and maximum number of solutions that a system of simultaneous linear equations can have, if it is consistent system?****Solution.** One and infinite

**Question. Give an equation of a line which passes through the origin.****Solution.** y = mx

**Question. For what value of k, will the equations 4x + my = 8 and 3x – 5y + 7 = 0 represent parallel lines?****Solution.** k = -20/3

**Question. If a system of equation is inconsistent, then what type of graph the equations will have?****Solution.** parallel lines

**Question. If x = – y, x = – 3, and x-axis form a triangle as shown, find the co-ordinates of the three vertices of the triangle.**

**Solution.** (0, 0), (–3, 0), (–3, 3)

**Question. For what value of k, the equations kx + 3y = k – 3 and 12 x + ky = k has infinitely many solutions?****Solution.** 6

**Question. What is the area of the triangle formed by the lines 2x + y = 6, 2x – y + 2 = 0 and x-axis in the figure given below ? **

**Solution.** 8 sq. units

**Question. Given a linear equation 3x – 5y = 15. Write another linear equation in two variables, such that the geometric representation of the pair so formed is coincident lines.****Solution.** 6x – 10y = 30

**Question. For what value of k, will the equations 2x – y + 8 = 0 and 4x – ky + 16 = 0 represent coincident lines?****Solution.** k = 2

**Question. What are the points of intersection of the line x/a + y/b + 3 = 0 with x-axis and with y-axis?****Solution.** (–3a, 0), (0, –3b)

**Question. For what value of k, will the equations 2x + ky = 7 and 3x + 9y = 13 may have a unique solution?****Solution.** All real numbers except 6.

**Question. Find a point on y-axis satisfying 3x – 4y = 12.****Solution.** (0, –3)

**Question. Use a single graph paper and draw the graph of the following equations : ****2y – x = 8; 5y – x = 14; y – 2x = 1.****Obtain the vertices of the triangle so obtained.****Solution.** For equation 2y – x = 8

We have, 2y – x = 8 ⇒ x = 2y – 8

When y = 2, we have x = 2 (2) – 8 = 4 – 8 = – 4

when y = 3, we have x = 2(3) – 8 = 6 – 8 = – 2

Thus, we have the following table :

For equation 5y – x = 14

We have, 5y – x = 14 ⇒ x = 5y – 14

when y = 3, we have x = 5(3) – 14 = 15 – 14 = 1

when y = 4, we have x = 5 (4) – 14 = 20 – 14 = 6

Thus, we have the following table :

For equation y – 2x = 1

We have, y – 2x = 1 ⇒ y = 2x + 1

when x = – 1, we have y = 2 (–1) + 1 = –2 + 1 = – 1

when x = 0, we have y = 2 (0) + 1 = 0 + 1 = 1

Thus, we have the following table :

The graph for the given equations is shown below :

From graph, we observe that the vertices of the triangle are (–4, 2), (1, 3) and (2, 5).

**Question. Find the values of a and b so that the following system of linear equations has an infinite number of solutions : 2x – 3y = 7 ****(a + b)x – (a + b – 3) y = 4a + b****Solution.**

Taking Ist two terms :

Taking last two terms :

From equation (1) we have a = – 6 – b

Using the value of a in equation (2), we get

5(–6 – b) – 4b = -21

⇒ –30 – 5b – 4b = – 21 ⇒ –9b ⇒ 9 ⇒ b = – 1

and a = – 6 – (–1) = – 6 + 1= – 5

Hence, a = – 5 and b = – 1 Ans.

**Question. Use the method of substitution to solve the following system of equations : **

**Solution.**

⇒ 2x + 3y = -10 …(2)

Now, we will solve equation (1) and (2) by substitution method.

From (1), we get y = 8 – 12 x.

Now, substituting the value of y in eqn. (2), we get

2x + 3 (8 – 12x) = –10

⇒ 2x + 24 – 36 x = –10

⇒ – 34x = – 34 ⇒ x = 1

Now, substitute x = 1 in y = 8 – 12 x, we get

y = 8 – 12 (1) = 8 – 12 = – 4

Thus x = 1, y = – 4 is the required solution

**Question. Solve for x and y : **

**Solution.**

**Question. Draw the graph of linear equation 2x + 3y = 7. ****Solution.** 2x + 3y = 7 ⇒ 3y = 7 – 2x ⇒ y = 7 – 2x /3

Give atleast two suitable values to x to find the corresponding value of y.

Representing it in a tabular form, we get

We now plot the points (2, 1) and (5, –1) on graph paper to obtain a straight line.

**Question. Aftab tells his daughter, ‘‘Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be. Represent this situation algebraically and graphically. ****Solution.** Let the present age of the daughter = x years.

7 years ago daughter’s age = (x – 7) years

3 years from now, daughter’s age = (x + 3) years

Let the present age of father = y years

7 years ago father’s age = ( y – 7) years

3 years from now, fathers age = (y + 3) years.

According to given question, two algebraic equations are :

y – 7 = 7 (x – 7) ⇒ y = 7x – 42 …(1)

and y + 3 = 3 (x + 3) ⇒ y = 3x + 6 …(2)

Required table for y = 7x – 42

and, required table for y = 3x + 6 is

From graph, we observe that :

Present age of father is 42 years and, present age of his daughter is 12 years.

**Question. The sum of a two-digit number and the number obtained by reversing the order of digits is 165. If the digits differ by 3, find the number. ****Solution.** Let the unit digit be x and tens digit be y.

Then, number = 10 y + x.

Number obtained by reversing the order of the digits = 10x + y.

According to the given question, we have

(10y + x) + (10 x + y) = 165 …(1)

and, x – y = 3 …(2)

OR

y – x = 3 …(3)

From eqn. (1), we get

11x + 11y = 165 ⇒ x + y = 15 …(4)

Solving eqn. (2) and (4) together, we get

x = 9, y = 6

Solving eqn. (3) and (4) together, we get

x = 6, y = 9

Substituting the values of x and y, for the number 10 y + x, we get number = 69 or 96. Ans.

**Question. Draw the graphs of the equations : ****x – y + 1 = 0; 3x + 2y – 12 = 0.****Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis and shade the triangular region. Also, find area of this triangular region.****Solution.** For equation x – y + 1 = 0

We have, x – y + 1 = 0 ⇒ y = x + 1

when, x = 0, we have y = 0 + 1 = 1

when, x = –1, we have y = –1 + 1 = 0

Thus, we have the following table :

For equation 3x + 2y – 12 = 0

we have, 3x + 2y – 12 = 0 ⇒ y = 12 – 3x/2

when x = 0, we have y = 12 – 3(0)/2 = 12/2 = 6

when x = 4, we have y = 12 – 3(4)/2 = 12 – 12 /2 = 0

Thus, we have the following table :

The graph for the given equations is shown below.

From graph, we observe that the vertices of ⇒ ABC are A(2, 3), B(4, 0) and C(–1, 0).

Also, Area of ABC = 1/2 x 5 x 3 sq. units = 7.5 sq. units.

**Question. The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle. ****Solution.** Let the length and breadth of the rectangle be x and y units respectively.

Then, Area = xy sq. units

If the length is reduced by 5 units and breadth is increased by 3 units, then area is reduced by 9 sq. units.

∴ xy – 9 = (x – 5) (y + 3)

⇒ xy – 9 = xy + 3x – 5y – 15

⇒ 3x – 5y = 6 …(1)

When the length is increased by 3 units and breadth by 2 units, the area is increased by 67 sq.

units.

⇒ xy + 67 = (x + 3) (y + 2)

⇒ xy + 67 = xy + 2x + 3y + 6

⇒ 2x + 3y = 61 …(2)

Solving Eq. (1) and eq. (2), we get

Hence, the length and breadth of the rectangle are 17 units and 9 units respectively.

**Question. A fraction is such that if the numerator is multiplied by 3 and the denominator is reduced by 3, ****we get 18/11 , but if the numerator is increased by 8 and the denominator is doubled, we get 2/5 .Find the fraction.****Solution.** Let the fraction be x/y

Then, according to given equestion, we have :

3x/y-3 = 18/11 and x + 8/2y = 2/5

⇒ 11x = 6y – 18 and 5x + 40 = 4y

⇒ 11x – 6y = – 18 …(1)

and 5x – 4y = – 40 …(2)

multiplying eqn. (1) by 2 and eqn. (2) by 3, we get :

22x – 12 y = – 36 …(3)

15x – 12y = – 120 …(4)

Subtracting eqn. (4) from eqn. (3), we get

7x = 84 ⇒ x = 12

Substituting x = 12 in eqn. (1), we get

11(12) – 6y = – 18

⇒ –6y = – 150 ⇒ y = 25.

Hence, the required fraction is 12/35 . Ans.

**Question. Solve the following system of linear equations by elimination method (equating the coefficients). ****6 (ax + by) = 3a + 2b****6 (bx – ay) = 3b – 2a ****Solution.** Given equations :

6 (ax + by) = 3a + 2b …(1)

6 (bx – ay) = 3b – 2a …(2)

multiplying eqn. (1) by a and equation (2) by b, and adding, we get

**Question. Solve the following system of linear equations by using the method of cross-multiplication : ****ax + by = a – b****bx – ay = a + b ****Solution.** The given system of equations may be written as

ax + by – (a – b) = 0

bx – ay – (a + b) = 0

Using cross-multiplication method, we get

**Question. A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car. ****Solution.** Let the speed of the train be x km/h and that of the car be y km/hr.

Case I : When he travels 250 km by train and rest by car :

In this case, we have

Time taken by the man to travel 250 km by train = 250/x hrs.

Time taken by the man to travel (370 – 250) km = 120 km by car = 120/y hrs.

**Question. A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs. 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs. 1180 as hostel charges.****Find the fixed charges and the cost of food per day. ****Solution.** Let the fixed charges be Rs. x and charges per day be Rs. y.

According to given question,

x + 20 y = 1000 …(1)

x + 26 y = 1180 …(2)

Subtracting eq. (1) from eqn. (2), we get

6y = 180 ⇒ y = 30

from eqn. (1), x + 20(30) = 1000

⇒ x = 1000 – 600 = 400

∴ Fixed charges = Rs. 400 and cost of food per day = Rs. 30.

**Question. 2 tables and 3 chairs together cost Rs. 2000 whereas 3 tables and 2 chairs together cost Rs. 2500. Find the total cost of 1 table and 5 chairs.****Solution.** Let the cost of a table be Rs. x and that of a chair be Rs. y. According to given question,

2x + 3y = 2000 …(1)

and 3x + 2y = 2500 …(2)

Adding eqn. (1) and (2), we get

5x + 5y = 4500 ⇒ x + y = 900 …(3)

Subtracting eqn. (1) from eqn. (2), we get

x – y = 500 …(4)

Adding eqn. (3) and eqn. (4), we get

2x = 1400 ⇒ x = 700

Using x = 700 in eqn. (3), we get

700 + y = 900 ⇒ y = 200

⇒ Cost of 1 table = Rs. 700 and cost of 1 chair = Rs. 200.

Hence, cost of 1 table and 5 chairs

= Rs. (x + 5y) = Rs. 700 + 5 (200) = Rs. 1700 Ans.

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