Linear Equations

CBSE Class 10 — Maths — Previous Year Questions

Q1.MCQ1 mark(2020)Easy

The value of $k$ for which the system of equations $x+y-4=0$ and $2x+ky=3$ has no solution, is:

Q2.MCQ1 mark(2020)Easy

The value of $k$ for which the system of linear equations $x+2y=3$, $5x+ky+7=0$ is inconsistent is

Q3.MCQ1 mark(2020)Medium

The value of $k$, for which the pair of linear equations $kx+y=k^2$ and $x+ky=1$ have infinitely many solutions is

Q4.MCQ1 mark(2020)Easy

The pair of linear equations $y=0$ and $y=-6$ has

Q5.MCQ1 mark(2020)Easy

The pair of linear equations $\frac{3x}{2}+\frac{5y}{3}=7$ and $9x+10y=14$ is

Q6.MCQ1 mark(2023)Medium

The value of $k$ for which the pair of equations $kx=y+2$ and $6x=2y+3$ has infinitely many solutions,
(A) is $k=3$
(B) does not exist
(C) is $k=-3$
(D) is $k=4$

Q7.Assertion-Reason1 mark(2023)Medium

Assertion (A): Point $P(0,2)$ is the point of intersection of y-axis and the line $3x+2y=4$.

Reason (R): The distance of point $P(0,2)$ from x-axis is 2 units.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true but Reason (R) is false.
(D) Assertion (A) is false but Reason (R) is true.

Q8.MCQ1 mark(2023)Easy

If the pair of equations $3x-y+8=0$ and $6x-ry+16=0$ represent coincident lines, then the value of 'r' is:
(a) $-\frac{1}{2}$
(b) $\frac{1}{2}$
(c) $-2$
(d) $2$

Q9.MCQ1 mark(2023)Easy

The pair of equations $x=a$ and $y=b$ graphically represents lines which are:
(a) parallel
(b) intersecting at $(b,a)$
(c) coincident
(d) intersecting at $(a,b)$

Q10.MCQ1 mark(2023)Easy

The pair of linear equations $2x=5y+6$ and $15y=6x-18$ represents two lines which are:

Q11.Case-Study1 mark(2023)Easy

Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹$x$ per student and Cricket ₹$y$ per student. School 'P' decided to award a total of ₹9,500 for the two games to 5 and 4 students respectively; while school 'Q' decided to award ₹7,370 for the two games to 4 and 3 students respectively.

Based on the above information, answer the following questions:

(i) Represent the following information algebraically (in terms of $x$ and $y$).

Q12.Case-Study1 mark(2023)Easy

(iii) What will be the total prize amount if there are 2 students each from two games?

Q13.MCQ1 mark(2023)Easy

The pair of equations $ax+2y=9$ and $3x+by=18$ represent parallel lines, where $a$, $b$ are integers, if:

Q14.MCQ1 mark(2023)Easy

The point of intersection of the line represented by $3x-y=3$ and $y$-axis is given by

Q15.MCQ1 mark(2023)Easy

$3$ chairs and $1$ table cost \u20b9 $900$; whereas $5$ chairs and $3$ tables cost \u20b9 $2,100$. If the cost of $1$ chair is \u20b9 $x$ and the cost of $1$ table is \u20b9 $y$, then the situation can be represented algebraically as

Q16.Case-Study1 mark(2023)Easy

A coaching institute of Mathematics conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, there are $20$ poor and $5$ rich children, whereas in batch II, there are $5$ poor and $25$ rich children. The total monthly collection of fees from batch I is \u20b9 $9000$ and from batch II is \u20b9 $26,000$. Assume that each poor child pays \u20b9 $x$ per month and each rich child pays \u20b9 $y$ per month.

Based on the above information, answer the following questions:

(i) Represent the information given above in terms of $x$ and $y$.

Q17.Case-Study1 mark(2023)Easy

(iii) If there are $10$ poor and $20$ rich children in batch II, what is the total monthly collection of fees from batch II?

Q18.MCQ1 mark(2024)Easy

In the given figure, graphs of two linear equations are shown. The pair of these linear equations is:

Q19.MCQ1 mark(2024)Easy

The value of $k$ for which the system of equations $3x-y+8=0$ and $6x-ky+16=0$ has infinitely many solutions, is

Q20.MCQ1 mark(2024)Easy

If $ax+by=a^2-b^2$ and $bx+ay=0$, then the value of $x+y$ is:

Q21.MCQ1 mark(2024)Easy

Two lines are given to be parallel. The equation of one of these lines is $5x-3y=2$. The equation of the second line can be:

Q22.MCQ1 mark(2024)Easy

Which out of the following type of straight lines will be represented by the system of equations $3x+4y=5$ and $6x+8y=7$?

Q23.MCQ1 mark(2025)Easy

If $x=1$ and $y=2$ is a solution of the pair of linear equations $2x-3y+a=0$ and $2x+3y-b=0$, then:

Q24.MCQ1 mark(2025)Easy

The line represented by the equation $x-y=0$ is:

Q25.Assertion-Reason1 mark(2025)Medium

Assertion (A): The pair of linear equations $px+3y+50=0$ and $2x+6y+118=0$ will have infinitely many solutions if $p=1$.

Reason (R): If the pair of linear equations $px+3y+19=0$ and $2x+6y+157=0$ has a unique solution, then $p\ne 1$.

Q26.MCQ1 mark(2025)Easy

The value of '$k$' for which the system of linear equations $6x+y=3k$ and $36x+6y=3$ has infinitely many solutions is:

Q27.MCQ1 mark(2025)Medium

The value of '$p$' for which the equations $px+3y=p-3$, $12x+py=p$ has infinitely many solutions is:

Q28.VSA2 marks(2023)Easy

Solve the pair of equations $x=3$ and $y=-4$ graphically.

Q29.VSA2 marks(2023)Easy

Using graphical method, find whether following system of linear equations is consistent or not:
$x=0$ and $y=-7$

Q30.Case-Study2 marks(2023)Medium

(ii) (a) What is the prize amount for hockey?

Q31.Case-Study2 marks(2023)Medium

(ii) (b) Prize amount on which game is more and by how much?

Q32.Case-Study2 marks(2023)Medium

(ii) Find the monthly fee paid by a poor child.

Q33.Case-Study2 marks(2023)Medium

(ii) Find the difference in the monthly fee paid by a poor child and a rich child.

Q34.VSA2 marks(2024)Easy

Solve the following system of linear equations:
$7x-2y=5$ and $8x+7y=15$
and verify your answer.

Q35.VSA2 marks(2025)Medium

Solve the following system of equations algebraically:
$30x+44y=10$; $40x+55y=13$

Q36.SA3 marks(2020)Medium

Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are given by $2y-x=8$, $5y-x=14$ and $y-2x=1$.

Q37.SA3 marks(2020)Easy

If $2x+y=23$ and $4x-y=19$, find the value of $(5y-2x)$ and $\left(\frac{y}{x}-2\right)$.

Q38.SA3 marks(2020)Medium

Solve the pair of equations:
$$\frac{2}{x}+\frac{3}{y}=11,\frac{5}{x}-\frac{4}{y}=-7$$
Hence, find the value of $5x-3y$.

Q39.SA3 marks(2020)Medium

Taxi charges in a city consist of fixed charges and the remaining charges depend upon the distance travelled. For a journey of 10 km, the charge paid is ₹75 and for a journey of 15 km, the charge paid is ₹110. Find the fixed charge and charges per km. Hence, find the charge of covering a distance of 35 km.

Q40.SA3 marks(2020)Medium

A fraction becomes $\frac{1}{3}$ when $1$ is subtracted from the numerator and it becomes $\frac{1}{4}$ when $8$ is added to its denominator. Find the fraction.

Q41.SA3 marks(2020)Medium

The present age of a father is three years more than three times the age of his son. Three years hence the father's age will be $10$ years more than twice the age of the son. Determine their present ages.

Q42.SA3 marks(2023)Easy

Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find the numbers.

Q43.SA3 marks(2023)Medium

If the system of linear equations $2x+3y=7$ and $2ax+(a+b)y=28$ have infinite number of solutions, then find the values of 'a' and 'b'.

Q44.SA3 marks(2023)Medium

If $217x+131y=913$ and $131x+217y=827$, then solve the equations for the values of $x$ and $y$.

Q45.SA3 marks(2023)Medium

Two people are $16$ km apart on a straight road. They start walking at the same time. If they walk towards each other with different speeds, they will meet in $2$ hours. Had they walked in the same direction with same speeds as before, they would have met in $8$ hours. Find their walking speeds.

Q46.SA3 marks(2024)Medium

Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old are Rashmi and Nazma now?

Q47.SA3 marks(2024)Easy

Rehana went to a bank to withdraw ₹$2,000$. She asked the cashier to give her ₹$50$ and ₹$100$ notes only. Rehana got $25$ notes in all. Find how many notes of ₹$50$ and ₹$100$ did she receive.

Q48.SA3 marks(2025)Medium

The monthly incomes of two persons are in the ratio 9 : 7 and their monthly expenditures are in the ratio 4 : 3. If each saved $₹5,000$, express the given situation algebraically as a system of linear equations in two variables. Hence, find their respective monthly incomes.

Q49.LA4 marks(2020)Medium

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?

Q50.LA4 marks(2020)Medium

A train covered a certain distance at a uniform speed. If the train would have been 6 km/hr faster, it would have taken 4 hours less than the scheduled time and if the train were slower by 6 km/hr, it would have taken 6 hrs. more than the scheduled time. Find the length of the journey.

Q51.Case-Study4 marks(2024)Medium

Easel World is one of India's largest amusement parks that offers a diverse range of thrilling rides, water attractions and entertainment options for visitors of all ages. The park is known for its iconic "Water Kingdom" section, making it a popular destination for family outings and fun-filled adventures. The ticket charges for the park are $\text{rupee}150$ per child and $\text{rupee}250$ per adult.

On a day, the cashier of the park found that 300 tickets were sold and an amount of $\text{rupee}55,000$ was collected.

Based on the above, answer the following questions:
(i) If the number of children visited be $x$ and the number of adults visited be $y$, then write the given situation algebraically.
(ii) (a) How many children visited the amusement park that day?
OR
(ii) (b) How many adults visited the amusement park that day?
(iii) How much amount will be collected if 250 children and 100 adults visit the amusement park?

Q52.Case-Study4 marks(2024)Medium

Easel World is one of India's largest amusement parks that offers a diverse range of thrilling rides, water attractions and entertainment options for visitors of all ages. The park is known for its iconic "Water Kingdom" section, making it a popular destination for family outings and fun-filled adventures. The ticket charges for the park are $\text{rupee}150$ per child and $\text{rupee}250$ per adult.

On a day, the cashier of the park found that 300 tickets were sold and an amount of $\text{rupee}55,000$ was collected.

Based on the above, answer the following questions:
(i) If the number of children visited be $x$ and the number of adults visited be $y$, then write the given situation algebraically.
(ii) (b) How many adults visited the amusement park that day?
(iii) How much amount will be collected if 250 children and 100 adults visit the amusement park?

Q53.Case-Study4 marks(2025)Medium

**Case Study - 1**

A school is organizing a grand cultural event to show the talent of its students. To accommodate the guests, the school plans to rent chairs and tables from a food supplier. It finds that rent for each chair is Rs 10 and for each table is Rs 200. The school spends Rs 30,000 for renting the total number of items (chairs and tables) rented are 300.

If the school rents '$x$' chairs and '$y$' tables, answer the following questions:

(i) Write down the pair of linear equations representing the given information.

(ii) (a) Find the number of chairs and number of tables rented by the school.

**OR**

(ii) (b) If the school wants to spend a maximum of Rs 27,000 on 300 items (tables and chairs), then find the number of chairs and tables it can rent.

(iii) What is maximum number of tables that can be rented in Rs 30,000 if no chairs are rented?

Q54.LA5 marks(2024)Medium

Using graphical method, solve the following system of equations:
$$3x+y+4=0$$
$$3x-y+2=0$$

Q55.LA5 marks(2024)Medium

Tara scored $40$ marks in a test, getting $3$ 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 wrong answer, then Tara would have scored $50$ marks. Assuming that Tara attempted all questions, find the total number of questions in the test.

Q56.LA5 marks(2025)Medium

Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received ₹1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received ₹20 more as annual interest. How much money did he invest in each scheme?