Are these practice questions for Pair of Linear Equations in Two Variables based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 10 Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Pair of Linear Equations in Two Variables free?

Yes. Every chapter includes free practice questions for Class 10 students to get you started.

What types of questions are included?

The practice set for Pair of Linear Equations in Two Variables mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Pair of Linear Equations in Two Variables for full coverage.

How many practice questions are there for Pair of Linear Equations in Two Variables?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Pair of Linear Equations in Two Variables Class 10 Mathematics Practice Questions with Answers

easy

Examine if

x=2, y=1

is a solution for the system of linear equations

2x + 3y = 7

and

4x - y = 5

.

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easy

Formulate a linear equation in two variables that is parallel to the line

2x - 5y + 7 = 0

and passes through the point

(5, 3)

.

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easy

Analyze if the pair of linear equations

2x - 3y = 7

and

4x - 6y = 15

is consistent or inconsistent.

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easy

The sum of two numbers is 35 and their difference is 13. Formulate a pair of linear equations to represent this situation.

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easy

Analyze the pair of linear equations

x + 2y = 4

and

2x + 4y = 12

to determine if their graphical representation will be intersecting lines, parallel lines, or coincident lines without drawing the graph.

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easy

State the condition for a pair of linear equations

a_1x + b_1y + c_1 = 0

and

a_2x + b_2y + c_2 = 0

to have a unique solution.

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easy

Name two algebraic methods for solving a pair of linear equations.

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easy

Define a consistent pair of linear equations.

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easy

Explain the difference between a consistent and an inconsistent pair of linear equations using their graphical representations.

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easy

What is the graphical representation of a pair of linear equations that has no solution?

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easy

Solve the pair of equations

x + y = 7

and

x - y = 1

.

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easy

Formulate a system of linear equations for the following scenario and then solve it: The sum of the present ages of a father and his son is 60 years. Six years ago, the father's age was five times the age of the son.

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medium

A fraction becomes

\frac{1}{3}

when 1 is subtracted from the numerator. It becomes

\frac{1}{4}

when 8 is added to its denominator. Calculate the fraction.

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medium

Summarize the three possible graphical representations for a pair of linear equations in two variables. For each case, describe the corresponding algebraic condition based on the ratios of their coefficients (

\frac{a_1}{a_2}

,

\frac{b_1}{b_2}

,

\frac{c_1}{c_2}

) and state the nature of their solutions (unique, infinitely many, or no solution).

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medium

Propose a system of two linear equations whose solution is the point

(-2, 5)

. Then, create a third linear equation that also passes through this point, such that it forms a dependent (coincident) system with one of the original equations. Justify your creations.

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medium

A student solved the system

x - 2y = 8

and

3x - 6y = 16

and concluded there are infinitely many solutions because

3(x - 2y) = 3(8)

gives

3x - 6y = 24

, which is "close" to the second equation. Critique the student's reasoning and provide the correct algebraic interpretation.

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medium

For the pair of equations

2x + 3y - 7 = 0

and

4x + 6y - 14 = 0

, compare the ratios

\frac{a_1}{a_2}

,

\frac{b_1}{b_2}

, and

\frac{c_1}{c_2}

and state the type of solution they have.

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medium

Design a word problem involving the cost of two items, for which the corresponding pair of linear equations would be

2x + 5y = 120

and

4x + 10y = 200

. Justify why the scenario described in your problem results in no possible solution.

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medium

For what value of

k

does the pair of equations

x - 2y = 3

and

3x + ky = 1

have a unique solution?

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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 ten years more than twice the age of the son. Calculate their present ages.

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medium

Identify if the pair of equations

x + y = 5

and

2x + 2y = 10

represents intersecting, parallel, or coincident lines.

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medium

The sum of a two-digit number and the number obtained by reversing its digits is 99. If the digits differ by 3, calculate the number. (Two possible answers exist, find both).

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medium

Solve the following pair of linear equations using the substitution method:

3x + 2y = 11

and

2x + 3y = 4

.

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medium

Critique the following statement: "For a pair of linear equations to be consistent, the lines representing them must intersect at a single point."

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medium

Design a pair of linear equations whose graphical representation results in intersecting lines, with the point of intersection being the origin

(0, 0)

. Justify your design.

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medium

Design a problem about a monthly mobile phone plan. The plan must involve a fixed monthly charge and a per-minute rate for extra talk time. The problem should require a user to formulate and solve a system of linear equations based on two different monthly bills. Finally, evaluate which of two given plans is more economical for a specific usage.

Problem Statement: A mobile phone company offers a plan with a fixed monthly charge and an additional charge for each extra minute of talk time. In April, a customer used 150 extra minutes and the total bill was ₹450. In May, the customer used 220 extra minutes and the total bill was ₹555. (a) Formulate a pair of linear equations to find the fixed monthly charge and the charge per extra minute. (b) Solve the system to find these charges. (c) The company offers a second plan with a fixed charge of ₹300 and a rate of ₹1.2 per extra minute. Evaluate which of the two plans is more economical for a usage of 300 extra minutes per month and justify your answer.

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medium

Describe the relationship between the coefficients of a pair of linear equations (

a_1x + b_1y + c_1 = 0

and

a_2x + b_2y + c_2 = 0

) when their graphs are coincident lines.

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medium

A student claims that the system

y = 3x + 5

and

y = -2x + 5

represents coincident lines because the constant term is the same in both equations. Evaluate this claim.

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medium

A taxi service in a city has a fixed charge along with a charge for the distance covered. For a journey of 12 km, the charge paid is ₹125. For a journey of 20 km, the charge paid is ₹197. Calculate the fixed charge and the charge per km. How much will a person have to pay for travelling a distance of 30 km?

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medium

Identify whether the pair of linear equations

4x - y = 5

and

8x - 2y = 9

is consistent or inconsistent. Explain how you know without solving them.

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medium

Summarize the steps of the substitution method for solving a pair of linear equations.

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medium

Solve for

x

and

y

:

\frac{x}{2} + \frac{y}{3} = 4

and

\frac{x}{3} + \frac{y}{2} = \frac{19}{6}

.

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medium

Explain the elimination method step-by-step. Describe what happens in this method if the pair of equations is (a) dependent (infinitely many solutions) and (b) inconsistent (no solution).

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hard

Solve the following pair of linear equations using the elimination method:

4x + 3y = 25

and

3x - 4y = -4

.

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hard

Calculate the values of

a

and

b

for which the following pair of linear equations has infinitely many solutions:

2x + 3y = 7

and

(a-b)x + (a+b)y = 3a+b-2

.

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hard

Given the equation

3x - 2y = 6

, write another linear equation in two variables such that the pair represents parallel lines. Explain your reasoning.

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hard

Formulate and solve a system of linear equations for the following problem: A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

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hard

Explain what it means algebraically when the substitution method results in a false statement like

0 = 5

.

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hard

The sum of a two-digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the original number, the new number is 4 more than 5 times the sum of the digits of the original number. Formulate a pair of linear equations for this problem and find the number. Justify that your solution is unique.

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hard

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. Calculate the dimensions of the rectangle.

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hard

Evaluate the statement: "A system of linear equations

a_1x + b_1y + c_1 = 0

and

a_2x + b_2y + c_2 = 0

will always have a unique solution, unless the lines are parallel." Critique this statement and then prove that if

\frac{a_1}{a_2} = \frac{b_1}{b_2}

, the system cannot have a unique solution.

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hard

Consider the general form of a pair of linear equations:

a_1x + b_1y + c_1 = 0

and

a_2x + b_2y + c_2 = 0

. Explain the geometric meaning of each of the following conditions: (i)

\frac{a_1}{a_2} \neq \frac{b_1}{b_2}

(ii)

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

(iii)

\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}

Also, relate each condition to the number of solutions the system has.

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hard

Critique the graphical method as a universal tool for solving linear equations. Formulate a specific pair of linear equations for which this method would be impractical and justify your choice.

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hard

For the pair of equations

2x + 3y = 7

and

(k-1)x + (k+2)y = 3k

, justify the value of

k

for which the pair will have infinitely many solutions.

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hard

Justify the choice of the elimination method over the substitution method for solving the system:

7x + 11y = 29

and

11x + 7y = 25

.

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Pair of Linear Equations in Two VariablesClass 10 Mathematics Practice Questions