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Linear Equations in Two Variables Class 10 Important Questio...

Linear Equations in Two Variables Class 10 Important Questions - Free PDF

Author at PW

August 19, 2025

Linear Equations in Two Variables Class 10 Important Questions are very useful if you're preparing for your upcoming board exams. This chapter explains how to solve equations with two variables and how to show these solutions on a graph. It also helps you understand the connection between algebra and geometry. Class 10 is an important year in every student’s life because board exams decide their next two years' course of studies. Therefore, giving your best at this stage becomes even more important.

Math is a subject where practice is important, and chapters like Linear Equations help in building strong problem-solving skills. So, by practicing Linear Equations in Two Variables Class 10 extra questions, you will get more practice, which can help you score better marks in exams. To explore Linear Equations in Two Variables Class 10 Important Questions and access the free PDF download link, keep reading.

Check Out: CBSE Class 10 Books

Linear Equations in Two Variables – Important Points

In order to solve class 10 Linear Equations in Two Variables extra questions, it is important that you first understand some of the key points from this chapter, as explained here:

What are Linear Equations?

A linear equation is an equation where the highest power of the variable is 1.
In other words, when we draw a linear equation on a graph, it always makes a straight line.
For example: y = 2x + 3

Linear Equations in Standard Form

A linear equation in two variables is generally written as ax + by + c = 0
Here, a, b, and c are constants, and x and y are variables.

Different Forms of Linear Equations

Slope-Intercept Form: y = mx + c (m is slope, c is intercept)
Point-Slope Form: y – y₁ = m(x – x₁)
Intercept Form: x/x₀ + y/y₀ = 1
Two-Point Form: y – y₁ = [(y₂ – y₁) / (x₂ – x₁)] (x – x₁)

Solving Linear Equations in One Variable

Take all terms with the variable to one side and constants to the other.
For example: x + 2 = 0 → x = –2

Check Out: CBSE Class 10 Revision Books

Solving Linear Equations in Two Variables

To find the values of x and y, we need to solve two equations together. The three common methods are:

(a) Substitution Method

In this method, we find the value of one variable from the first equation.
Then, we put (substitute) this value into the second equation.
This gives the value of the other variable.

For example: x + y = 5 → x = 5 – y
Put in the second equation: x – y = 1 → (5 – y) – y = 1 → y = 2, x = 3

(b) Elimination Method

In the Elimination Method, we add or subtract the two equations in a way that one variable gets cancelled.
Then we solve for the remaining variable.
After that, we put the value back into one of the equations to find the second variable.

For example: x + y = 5
x – y = 1
Add them: 2x = 6 → x = 3
Now put in first: 3 + y = 5 → y = 2

(c) Cross Multiplication Method

This method is used when equations are in standard form:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
The formula is:

x / (b₁c₂ – b₂c₁) = y / (c₁a₂ – c₂a₁) = 1 / (a₁b₂ – a₂b₁)

By solving this equation, we will be able to get the values of x and y.

Check Out: CBSE Class 10 Question Banks

Linear Equations In Two Variables Class 10 Important Questions

For Class 10 board exam preparation, practicing the right questions is very important. To help students, PW's mentors have prepared a set of exam-focused Linear Equations in Two Variables class 10 important questions. These questions cover all important concepts and common types of questions that are generally asked in exams.

By solving the following Linear Equations class 10 extra questions, you will not only revise the full chapter but also understand the exam pattern better.

Question: The given pair of linear equations x + 2y – 5 = 0 and –3x – 6y + 15 = 0 have:

(a) Exactly one solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No solution

Question: For what value of k will the equations x + 2y + 7 = 0, 2x+ ky + 14 = 0 represent coincident lines?

(a) 6
(b) 4
(c) 3
(d) 2

Question: If a pair of linear equations is consistent, then the lines represented by them are:

(a) parallel
(b) intersecting or coincident
(c) always coincident
(d) always intersecting

Question: Is the following system of linear equations consistent?

2x + 3y = 7
3x – y = 5

Question: The two circles represent the ordered pairs, (a, b), which are solutions of the respective equations. The circles are divided into 3 regions P, Q and R as shown.

Write one ordered pair each belonging to P, Q and R. Show your work.

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

(a) –2
(b) –3
(c) 3
(d) 2

Question: 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.

Linear Equations in Two Variables Class 10 Extra Questions with Solutions

Find out some of the Linear Equations in Two Variables Class 10 extra questions with solutions that are provided in the free PDF prepared by PW’s mentors. These extra questions are selected after checking past exam patterns and NCERT exercises, so you can practice what's needed for board exam preparation.

Direction: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as:

(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.

Q1. Assertion (A): If x = 3, y = 1 is the solution of the line 2x + y – q² – 3 = 0, then the value of q = ±2.
Reason (R): The solutions of the line will satisfy the equation of the line. (An)

Solution:
It is given that x = 3 and y = 1 is the solution of line 2x + y – q² – 3 = 0
⇒ 2 × 3 + 1 – q² – 3 = 0
⇒ 6 + 1 – 3 – q² = 0
⇒ 4 – q² = 0
⇒ q² = 4
⇒ q = ±2

Q2. Assertion (A): If x = 3, y = 1 is the solution of the line 2x + y – q² – 3 = 0, then the value of q = ±2.
Reason (R): The solutions of the line will satisfy the equation of the line. (Un)

Solution:
It is given that x = 3 and y = 1 is the solution of line 2x + y – q² – 3 = 0
⇒ 2 × 3 + 1 – q² – 3 = 0
⇒ 6 + 1 – 3 – q² = 0
⇒ 4 – q² = 0
⇒ q = ±2

Hence, both A and R are true, and R explains A.

Question. Aruna has only INR 1 and INR 2 coins with her. If the total number of coins is 50 and the total amount is INR 75, then the number of INR 1 and INR 2 coins are, respectively:
(a) 35 and 15
(b) 35 and 20
(c) 15 and 35
(d) 25 and 25

Solution:
Let the number of INR 1 coins = x
Let the number of INR 2 coins = y

According to the question,
x + y = 50 …(i)
x × 1 + 2 × y = 75 ⇒ x + 2y = 75 …(ii)

Subtracting (i) from (ii):
(x + 2y) – (x + y) = 75 – 50
y = 25

Substitute y = 25 in (i):
x + 25 = 50
x = 25

Thus, the number of INR 1 and INR 2 coins is 25 and 25, respectively.

Question: The father’s age is six times his son’s age. After four years, the age of the father will be four times his son’s age. Find the present ages of the son and the father, in years. (Ev)

(a) 8 and 32 (b) 5 and 30 (c) 6 and 36 (d) 3 and 24

Solution: Let the present age of the father and his son be x years and y years, respectively.

According to the question,
x = 6y …(i)
and (x + 4) = 4(y + 4)
⇒ x – 4y = 12 …(ii)

Substituting the value of x from the equation. (i) in eqn. (ii), we get
6y – 4y = 12
⇒ 2y = 12
⇒ y = 6

Now, x = 6y = 6 × 6 = 36 [from eqn. (i)]

Thus, the present ages of the father and son are 36 years and 6 years, respectively.

Question: 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.

Solution: Let the present age of son = x
And, the present age of father = 3x + 3

3 years later,
Age of son = x + 3
Age of father = 3x + 3 + 3 = 3x + 6 …(i)

According to the given condition,
Age of father = 10 + 2(x + 3)

∴ 3x + 6 = 10 + 2(x + 3)
⇒ 3x + 6 = 10 + 2x + 6
⇒ 3x – 2x = 10 + 6 – 6
⇒ x = 10

∴ Present age of son = 10 years
Present age of father = 3x + 3 = 3 × 10 + 3 = 33 years

To practice more such types of Linear Equations in Two Variables Class 10 important questions to boost your board exam preparation, download the free PDF from the link provided below.

Linear Equations In Two Variables Class 10 Important Questions PDF

Importance of Class 10 Linear Equations in Two Variables Extra Questions

Linear Equations in Two Variables Class 10 extra questions can be a game-changer for you. These questions help you in understanding the concepts deeply and let you apply the learned concepts and formulas to real problems. Here are some benefits of the Linear Equations class 10 extra questions:

Better Understanding of Concepts: Practicing extra questions makes the basics of linear equations easier to understand.
Good for Board Exams: Class 10 board exams generally include tricky and tough questions. So by solving Linear Equations in Two Variables extra questions, you get prepared to solve different types of questions.
Step-by-Step Practice: The solutions given with the linear equations in two variables class 10 important questions help you learn how to write answers in proper steps in exams.

Builds Confidence: The more questions you solve, the more confidence you get in this chapter.
Covers Important Patterns: Linear Equations in Two Variables extra questions include questions from the NCERT exemplar and previous years' Class 10th papers that are generally asked in exams.
Useful for Higher Classes: This chapter forms the base for algebra in Classes 11 and 12, so practicing as many linear equations in two variables extra questions as possible will help you later as well.

Tips to Prepare Maths for Class 10th Boards To Score Well

Maths is one of the most important subjects in Class 10 as it builds the base for higher studies and can also help you score well. With the right practice and regular revision, you can improve speed, accuracy, and confidence in solving questions. Here are some key tips that help you prepare maths for the Class 10th board exam:

Focus on the NCERT Book First

Most of the board exam questions are based on the NCERT. Go through every exercise and example carefully. Make sure to practice each question step by step. You can also add study resources like NCERT Exemplar Problems Mathematics by PW to deepen your understanding of important concepts.

Practice Important Chapters Regularly

Give your extra time and attention to topics like Real Numbers, Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmetic Progressions, Triangles, Coordinate Geometry, Introduction to Trigonometry, Circles, Statistics, and Probability.

Just reading is not enough, so practice plenty of questions like the Class 10 Linear Equations in Two Variables extra questions provided here.

Solve Sample Papers To Understand the Question Types

CBSE Class 10 Sample Papers give you exam-like practice. They help you improve time management by practicing with the chapters that are most important. PW CBSE Class 10th sample papers include:

Cheat Sheets for quick revision
100 Most Probable Questions for exam focus
CBSE Solved Papers as per the latest pattern
CBSE SQPs and APQs for practice
Sample Question Papers with detailed explanations

Revise with Question Banks

After completing the NCERT, use the CBSE Class 10th Question Banks by PW. They include solved past-year papers, concept maps, mind maps, and extra practice questions. This makes your preparation stronger and helps with last-minute revision.

Daily Practice of Formulas and Theorems

Write down all formulas and theorems in one notebook. Use CBSE Class 10 Mind Maps Book to save time, improve speed, and avoid silly mistakes in the exam.

Mock Tests for Real Exam Practice

Attempt mock tests in a time-limited condition. This will boost confidence and help you perform better under exam pressure.

Linear Equations In Two Variables Class 10 Important Questions FAQs

Q.1. Where can I get linear equations in two variables class 10 extra questions for practice?

Ans. You can download the Linear Equations in Two Variables Class 10 Important Questions PDF provided above. It is specially made for exam-focused practice and will help you revise this chapter better.

Q.2. Is linear equations in two variables an important chapter for the board exam?

Ans. Yes, Linear Equations is one of the important chapters for the Class 10 board exams. Every year, at least one or two questions are asked from it, which makes it a scoring topic.

Q.3. Is practicing Linear Equations in Two Variables Class 10 Important Questions enough for boards?

Ans. Practicing important questions is very important, as it gives you an idea of exam-type problems. But along with this, you should also complete NCERT exercises, examples, and sample papers to prepare fully.

Q.4. What are the methods for solving linear equations?

Ans. There are four main methods to solve linear equations in two variables. These are the substitution method, elimination method, cross-multiplication method, and graphical method.

Q.5. What is the real-life importance of the Linear Equations in Two Variables chapter?

Ans. This chapter is useful in many real-life situations. It helps you solve problems related to ages, money, cost, and price, and other cases where two unknown values are given.

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