# Chapter 3 Pair of Linear Equations in Two Variables Important Questions for CBSE Class 10 Maths Board Exams

**Important Questions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables**, which will help the students to prepare for the CBSE Class 10 maths Board exam 2022-23. It help the doing better in their maths paper. Extra questions of Chapter 3 Pair of Linear Equations in Two Variables given here which are based on the pattern of CBSE NCERT book. Students will learn about the entire syllabus and learn how to solve problems in preparation for the exams.

## Important Questions for Chapter 3 Pair of Linear Equations in Two Variables Class 10 Maths

### Pair of Linear Equations in Two Variables Class 10 Maths Important Questions Very Short Answer (1 Mark)

**1. Find whether the pair of linear equations y = 0 and y = -5 has no solution, unique solution or infinitely many solutions.**

**Solution**

**2. If , am = bl then find whether the pair of linear equations ax + by = c and lx + my = n has no**

**solution, unique solution or infinitely many solutions.**

**Solution**

**3. If , ad ≠bc then find whether the pair of linear equations ax + by = p and cx + dy = q has no solution,**

**unique solution or infinitely many solutions.**

**Solution**

**4. How many solutions does the pair of equations y = 0 and y = -5 have?**

**Solution**

**5. Two lines are given to be parallel. The equation of one of the lines is 4x + 3y = 14, then find the equation of the second line.**

**Solution**

The equation of one line is 4x + 3y = 14.

We know that if two lines a_{1}x + b_{1}y + c = 0 and a_{2}x + b_{2}y + c = 0 are parallel, then

**6. If ax + by = a ^{2} – b^{2} and bx + ay = 0, find the value of (x + y).**

Solution

**7. Calculate the area bounded by the line x + y = 10 and both the co-ordinate axes. (2012)**

**Solution**

Area of triangle

= 12 × base × corresponding altitude

= 12 × 10 × 10 = 50 cm^{2}

**8. If the equations kx - 2y = 3 and 3x + y = 5= represent two intersecting lines at unique point, then the value of k is ........... .**

**Solution**

For unique solution

**9. If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is .....**

**Solution**

inconsistent

**10. If a pair of linear equations has solution, either a unique or infinitely many, then it is said to be ......**

**Solution**

consistent

### Pair of Linear Equations in Two Variables Class 10 Maths Important Questions Short Answer-I (2 Marks)

**11. Solve by elimination:**

**3x – y – 72x + 5y + 1 = 0**

**Solution**

3x – y = 7 **…(i)**

2x + 5y = -1 -00

Multiplying equation (i) by 5 & (ii) by 1,

⇒ x = 2

Putting the value of x in (i), we have

3(2)-y = 7

⇒ 6 – 7 = y

∴ y = -1 ∴ x = 2, y = -1

**12. Find the value(s) of k so that the pair of equations ****x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.**

**Solution**

We have,

x + 2y - 5 = 0 ...(1)

and 3x + ky + 15 = 0 ...(2)

Comparing equation (1) with a_{1}x + b_{1}y + c_{1} = 0 and equation (2) with a_{2}x + b_{2}y + c_{2} = 0, we get

a_{1} = 1, a_{2} = 2, b_{1} = 2, b_{2} = k, c_{1} = -5, c_{2} = 15

Since, given equations have unique solution, So

Hence, for all values of k except 6, the given pair of equations have unique solution.

**13. If 2x + y = 23 and 4x - y = 19, find the value of (5y - 2x) and (y/x - 2).**

**Solution**

We have,

2x + y = 23 **...(1)**

4x - y = 19 **...(2)**

Adding equation (1) and (2), we have

6x = 42

⇒ x = 7

Substituting the value of x in equation (1), we get

14 + y =23

⇒ y = 23 - 14 = 9

Hence,

5y - 2x = 5×9 - 2×7

= 45 - 14

= 31

**14. Find whether the following pair of linear equations is consistent or inconsistent:**

3x + 2y = 8 6x – 4y = 9

3x + 2y = 8 6x – 4y = 9

**Solution**

Therefore, given pair of linear equations is consistent.

**15. Draw the graph of**

**2y = 4x – 6; 2x = y + 3 and determine whether this system of linear equations has a unique solution or not.**

**Solution**

Since both the lines coincide.

Therefore infinitely many solutions.

**16. Find whether the lines represented by 2x + y = 3 and 4x + 2y = 6 are parallel, coincident or intersecting.**

**Solution**

Here,

a_{1} = 2, b_{1} = 1, c_{1} = -3 and a_{2} = 4, b_{2} = 2, c_{2} = -6

**17. Find whether the following pair of linear equation is consistent or inconsistent:**

**3x + 2y= 8, 6x - 4y= 9**

**Solution**

Hence, the pair of linear equation is consistent.

**18. Is the system of linear equations 2x+ 3y - 9 = 0 and 4x+ 6y - 18 = 0 consistent? Justify your answer.**

**Solution**

For the equation, 2x+ 3y - 9 = 0 we have

a_{2} = 2, b_{1} = 3 and c_{1} = -9 and

for the equation, 4x+ 6y - 18 = 0 we have

a_{2} = 4, b_{2} = 6 and c_{2} = -18

### Pair of Linear Equations in Two Variables Class 10 Maths Important Questions Short Answer-II (3 Marks)

**19. Solve the following system of equations.**

**21/x + 47/y = 110, 47/x + 21/y = 162, x, y ≠ 0**

**Solution**

**20. A fraction becomes 1/3 when 2 is subtracted from the numerator and it becomes 1/2 when 1 is subtracted from the denominator. Find the fraction.**

**Solution**

**...(2)**

**21. Solve the following pair of equations for x and y:a ^{2}/x − b^{2}/y = 0; a^{2}b/x + b^{2}a/y = a + b, x ≠ 0; y ≠ 0**

**Solution**

**22. In the figure, ABCDE is a pentagon with BE||CD and BC||DE. BC is perpendicular to CD. AB = 5cm, AE = 5 cm, BE = 7 cm, BC = x - y and CD = x + y. If the perimeter of ABCDE is 27 cm. Find the value of x and y, given x, y ≠ 0.**

**Solution**

**...(1)**

**...(2)**

**23. Given the linear equation 2x + 3y - 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is :**

**(a) intersecting lines**

**(b) parallel lines**

**(c) coincident lines.**

**Solution**

**...(1)**

**...(2)**

**24. Determine the values of m and n so that the following system of linear equation have infinite number of solutions :**

**(2m - 1)x + 3y - 5 = 0**

**3x + (n-1)y - 2 = 0**

**Solution**

We have (2m -1)x + 3y - 5 = 0 **...(1)**

Here, a_{1} = 2m -1, b_{1} = 3, c_{1} = -5

3x + (n-1)y - 2 = 0 **...(2)**

Here, a_{2} = 3, b_{2} = (n-1), c_{2} = -2

For a pair of linear equations to have infinite number of solutions,

**25. Find the values of Î± and Î² for which the following pair of linear equations has infinite number of solutions :**

**2x + 3y = 7, 2Î±x + (Î±+Î²)y = 28**

**Solution**

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

**10/x+y + 2/x−y = 4; 15/x+y − 5/x−y = −2**

**x + y ≠ 0**

**x – y ≠ 0**

**Solution**

**27. Solve by elimination:**

**3x = y + 5**

**5x – y = 11**

**Solution**

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

**27x + 31y = 85;**

**31x + 27y = 89**

**Solution**

**29. Solve for x and y: x/a - y/b = 0;**

**ax + by = a ^{2}+ b^{2}**

**Solution**

Putting the value of x in (i), we get

b(a) – ay = 0

⇒ ba = ay

ba/a = y

∴ b = y

∴ x = a, y = b

**30. Find the two numbers whose sum is 75 and difference is 15.**

**Solution**

Let the two numbers be x and y.

According to the question,

x + y = 75 **…(i)**

∴ x – y = ±15 **…(ii)**

Solving (i) and (ii), we get

**31. Sum of the ages of a father and the son is 40 years. If**

**father’s age is three times that of his son, then find**

**their respective ages.**

**Solution**

**...(1)**

**...(2)**

**32. The sum of the digits of a two digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number. (2015)**

**Solution**

Let unit and tens digit be x and y.

∴ Original number = 1x + 10y **…(i)**

Reversed number = 10x + 1y

According to question,

x + y = 8

⇒ y = 8 – x **…(ii)**

Also, 1x + 10Oy – (10x + y) = 18

⇒ x + 10y – 10x – y = 18

⇒ 9y – 9x = 18

⇒ y – x = 2 **…[Dividing both sides by 9]**

⇒ 8 – x – x = 2 **…[From (ii)]**

⇒ 8 – 2 = 2x

⇒ 2x = 6

From (it), y = 8 – 3 = 5

From (i), Original number = 3 + 10(5) = 53

### Pair of Linear Equations in Two Variables Class 10 Maths Important Questions Long Answer (4 Marks)

**33. A man can row a boat downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find his speed of rowing in still water. Also find the speed of the stream.**

**Solution**

**34. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for four hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. How long would it take for each pipe to fill the pool separately?**

**Solution**

**35. Amit bought two pencils and three chocolates for ₹11 and Sumeet bought one pencil and two chocolates for ₹7. Represent this situation in the form of a pair of linear equations. Find the price of one pencil and that of one chocolate graphically.**

**Solution**

Let the price of one pencil = ₹x and the price of one chocolate = ₹y.

As per the Question,

Lines intersect at (1, 3).

∴ x = 1, y = 3

Therefore the price of one pencil = ₹1 and price of one chocolate = ₹3

**36. Draw the graphs of following equations:**

**2x – y = 1; x + 2y = 13**

**Find the solution of the equations from the graph and shade the triangular region formed by the lines and the y-axis. (2013)**

**Solution**

By plotting the points and joining them, the lines intersect at A(3,5).

∴ x = 3, y = 5

Here ∆ABC is the required triangle.

**37. For what value of k, which the following pair of linear ****equations have infinitely many solutions:**

**2x + 3y = 7 and (k+1)x + (2k-1)y = 4k + 1**

**Solution**

We have,

2x + 3y = 7

and (k+1)x + (2k-1)y = 4k + 1

Hence, the value of k is 5, for which the given equation have infinitely many solutions.

**38. The owner of a taxi company decides to run all the taxis on CNG fuel instead of petrol/diesel. The taxi charges in city comprises of fixed charges together with the charge for the distance covered. For a journey of 13 km, the charge paid is ₹129 and for a journey of 22 km, the charge paid is ₹210. What will a person have to pay for travelling a distance of 32 km?**

**Solution**

Let fixed charge be ₹x and the charge for the distance = ₹y per km

According to the Question,

For a journey of 13 km,

x + 13y = 129

⇒ x = 129 – 13y **…(i)**

For a journey of 22 km,

x + 22y = 210 **…(ii)**

⇒ 129 – 13y + 22y = 210 **…[From (i)]**

⇒ 9y = 210 – 129 = 81

⇒ 9y = 81

⇒ y = 9

From (i),

x = 129 – 13(9)

= 129 – 117 = 12

∴ Fixed charge, x = ₹12

∴ The charge for the distance, y = ₹9 per km

To pay for travelling a distance of 32 km

= x + 32y = 12 + 32(9) = 12 + 288 = ₹300