Pair of Linear Equations in Two Variables

Solving Linear Equations

Graphical Representation of Linear Equations

Consistency and Solutions of Linear Equations

Formulating Linear Equations from Word Problems

Algebraic Methods for Solving Equations

Interpreting Solutions of Linear Equations

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

Understand how to solve pairs of linear equations using substitution, elimination, and graphical methods.
Learn to represent linear equations graphically and interpret the intersection, parallelism, or coincidence of lines.
Determine the consistency of linear equations and the nature of their solutions (unique, none, or infinite).
Translate real-world scenarios into linear equations and solve them to find unknown values.
Use algebraic techniques such as substitution and elimination to solve linear equations.
Analyze the solutions of linear equations to understand their implications in real-world contexts.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter Notes

Solving Linear Equations

Linear Equations in Two Variables: Equations of the form $ax + by + c = 0$ .
Methods:
- Substitution Method: Solve one equation for one variable and substitute into the other.
- Elimination Method: Multiply equations to align coefficients and eliminate one variable by addition or subtraction.
- Graphical Method: Plot equations on a graph to find intersections.

Graphical Representation of Linear Equations

Intersection: Lines intersect at a point, indicating a unique solution.
Parallel Lines: No intersection, indicating no solution.
Coincident Lines: Lines overlap, indicating infinitely many solutions.

Consistency and Solutions of Linear Equations

Consistent Equations: Have at least one solution.
Inconsistent Equations: Have no solution.
Dependent Equations: Have infinitely many solutions.

Formulating Linear Equations from Word Problems

Translate scenarios into mathematical equations.
Example: "Five pencils and seven pens cost ₹50, while seven pencils and five pens cost ₹46."

Algebraic Methods for Solving Equations

Substitution: Solve for one variable and substitute.
Elimination: Align coefficients, add/subtract to eliminate a variable.

Interpreting Solutions of Linear Equations

Unique Solution: Indicates a specific solution point.
No Solution: Parallel lines, no intersection.
Infinite Solutions: Coincident lines, overlap completely.

Examples

Example 1: Solve $x + y = 5$ and $2x - 3y = 4$ using elimination.
Example 2: Graphically solve $x + 3y = 6$ and $2x - 3y = 12$ .
Example 3: Formulate and solve equations from word problems, e.g., age-related problems or cost calculations.

Verification

Always verify solutions by substituting back into the original equations to ensure correctness.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes & Exam Tips

Solving Linear Equations

Mistake: Students often make errors in arithmetic calculations when using substitution or elimination methods.
- Tip: Double-check each step of your calculations, especially when dealing with fractions or decimals.
Mistake: Incorrectly isolating variables in the substitution method.
- Tip: Always ensure that one variable is completely isolated before substituting it into the other equation.

Graphical Representation of Linear Equations

Mistake: Misreading the graph when the solution involves non-integral coordinates.
- Tip: Use algebraic methods to verify graphical solutions, especially when coordinates are not whole numbers.
Mistake: Failing to plot enough points to accurately draw the lines.
- Tip: Plot at least two points for each line and ensure they are correctly calculated to draw accurate graphs.

Consistency and Solutions of Linear Equations

Mistake: Confusing the conditions for consistency, inconsistency, and dependency.
- Tip: Remember:
  - Intersecting lines have a unique solution (consistent).
  - Parallel lines have no solution (inconsistent).
  - Coincident lines have infinitely many solutions (dependent and consistent).

Formulating Linear Equations from Word Problems

Mistake: Misinterpreting the problem statement leading to incorrect equations.
- Tip: Carefully translate each part of the word problem into mathematical expressions, and verify the logic before solving.

Algebraic Methods for Solving Equations

Mistake: Skipping steps in the elimination method, leading to errors.
- Tip: Follow each step methodically: equalize coefficients, eliminate one variable, solve for the other, and back-substitute.

Interpreting Solutions of Linear Equations

Mistake: Ignoring the context of the problem when interpreting solutions.
- Tip: Always relate the solution back to the original problem to ensure it makes sense in the given context.

By being aware of these common pitfalls and following the provided tips, students can improve their accuracy and efficiency in solving linear equations.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

Correct Answer: A

Solution:

Let the number of pants be

x

and the number of skirts be

y

. We have the equations:

y = 2x - 2

and

y = 4x - 4

. Solving these, we find

x = 1

and

y = 0

. Thus, Champa bought 1 pant.

Chapter Concept:

Formulating Linear Equations from Word Problems

Substitution Method

Elimination Method

Graphical Method

Matrix Method

Correct Answer: B

Solution:

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other.

Chapter Concept:

Algebraic Methods for Solving Equations

Unique solution

No solution

Infinitely many solutions

Cannot be determined

Correct Answer: C

Solution:

The second equation is a multiple of the first, indicating that the lines are coincident. Therefore, there are infinitely many solutions.

Chapter Concept:

Consistency and Solutions of Linear Equations

They have no solution.

They have a unique solution.

They have infinitely many solutions.

They form a triangle.

Correct Answer: C

Solution:

Dependent linear equations coincide, meaning they have infinitely many solutions.

Chapter Concept:

Graphical Representation of Linear Equations

They intersect at a single point.

They are parallel.

They coincide.

They form a right angle.

Correct Answer: C

Solution:

The second equation is a multiple of the first, indicating that the lines coincide, representing infinitely many solutions.

Chapter Concept:

Graphical Representation of Linear Equations

They intersect at a single point.

They are parallel.

They coincide.

They do not intersect.

Correct Answer: A

Solution:

The lines intersect at a single point, indicating a unique solution.

Chapter Concept:

Graphical Representation of Linear Equations

Unique solution

No solution

Infinitely many solutions

Dependent equations

Correct Answer: C

Solution:

The equations are coincident after manipulation, indicating infinitely many solutions.

Chapter Concept:

Interpreting Solutions of Linear Equations

\frac{3}{5}

\frac{4}{5}

\frac{5}{7}

\frac{7}{9}

Correct Answer: A

Solution:

Let the fraction be

\frac{x}{y}

. The equations are:

\frac{x+2}{y+2} = \frac{9}{11}

and

\frac{x+3}{y+3} = \frac{5}{6}

. Solving these, we find the fraction is

\frac{3}{5}

Chapter Concept:

Formulating Linear Equations from Word Problems

Unique solution

Infinitely many solutions

No solution

The equations are inconsistent

Correct Answer: C

Solution:

The equations are parallel lines, as shown by the ratios of coefficients, indicating no solution.

Chapter Concept:

Solving Linear Equations

Substitution Method

Elimination Method

Both A and B

None of the above

Correct Answer: C

Solution:

Both substitution and elimination methods are algebraic methods that can be used to solve linear equations when the graphical method is not convenient.

Chapter Concept:

Consistency and Solutions of Linear Equations

The equations have a unique solution.

The equations have no solution.

The equations have infinitely many solutions.

The equations are dependent.

Correct Answer: B

Solution:

Parallel lines indicate that the equations are inconsistent and have no solution.

Chapter Concept:

Solving Linear Equations

They intersect at one point.

They are parallel.

They are coincident.

They form a right angle.

Correct Answer: C

Solution:

The equations

3x + 2y = 5

and

6x + 4y = 10

are multiples of each other, indicating that the lines are coincident and have infinitely many solutions.

Chapter Concept:

Algebraic Methods for Solving Equations

Correct Answer: B

Solution:

Let the number of ₹ 50 notes be

x

and the number of ₹ 100 notes be

y

. We have the equations:

x + y = 25

and

50x + 100y = 2000

. Solving these, we find

x = 15

and

y = 10

. Therefore, Meena received 15 ₹ 50 notes.

Chapter Concept:

Solving Linear Equations

They have a unique solution.

They have no solution.

They have infinitely many solutions.

They are inconsistent.

Correct Answer: A

Solution:

The lines intersect at a single point, indicating a unique solution, making the pair of equations consistent.

Chapter Concept:

Interpreting Solutions of Linear Equations

Graphical method; solution is

x = 2

y = 3

Substitution method; solution is

x = 4

y = 2

Elimination method; solution is

x = 4

y = 2

None of the above

Correct Answer: C

Solution:

The elimination method is efficient because it allows us to eliminate one variable by adding or subtracting the equations. Solving, we find

x = 4

and

y = 2

Chapter Concept:

Solving Linear Equations

Unique solution

No solution

Infinitely many solutions

Exactly two solutions

Correct Answer: C

Solution:

When the lines are coincident, there are infinitely many solutions as each point on the line is a solution.

Chapter Concept:

Consistency and Solutions of Linear Equations

Intersecting lines

Parallel lines

Coincident lines

None of the above

Correct Answer: C

Solution:

Both equations are equivalent, meaning the lines are coincident and have infinitely many solutions.

Chapter Concept:

Consistency and Solutions of Linear Equations

The lines will intersect at a point.

The lines will be parallel.

The lines will coincide.

The lines will form a triangle with the x-axis.

Correct Answer: C

Solution:

The second equation is a multiple of the first, indicating that the lines will coincide.

Chapter Concept:

Consistency and Solutions of Linear Equations

₹ 9

₹ 12

₹ 15

₹ 18

Correct Answer: A

Solution:

Let the fixed charge be

f

and the additional charge per day be

c

. We have the equations:

f + 4c = 27

and

f + 2c = 21

. Solving these, we find

f = 9

and

c = 6

. Thus, the fixed charge is ₹ 9.

Chapter Concept:

Formulating Linear Equations from Word Problems

Unique solution

No solution

Infinitely many solutions

Dependent solution

Correct Answer: A

Solution:

The lines intersect at a single point, providing a unique solution.

Chapter Concept:

Graphical Representation of Linear Equations

Each pencil costs ₹2 and each eraser costs ₹1

Each pencil costs ₹1 and each eraser costs ₹2

The equations are dependent, and there are infinitely many solutions

The equations are inconsistent, and there is no solution

Correct Answer: C

Solution:

Both equations represent the same line, indicating they are dependent and consistent with infinitely many solutions.

Chapter Concept:

Solving Linear Equations

Multiply the second equation by 5

Multiply the second equation by 3

Add the two equations

Subtract the first equation from the second

Correct Answer: B

Solution:

Multiplying the second equation by 3 makes it identical to the first, indicating coincident lines with infinitely many solutions.

Chapter Concept:

Solving Linear Equations

2x + 3y = 9

and

4x + 6y = 18

x + 2y = 4

and

2x + 4y = 12

5x - 8y + 1 = 0

and

3x - \frac{24}{5}y + \frac{3}{5} = 0

x + 3y = 6

and

2x - 3y = 12

Correct Answer: A

Solution:

The equations

2x + 3y = 9

and

4x + 6y = 18

are multiples of each other, indicating coincident lines with infinitely many solutions.

Chapter Concept:

Graphical Representation of Linear Equations

Correct Answer: A

Solution:

Let the digits be

x

and

y

. The equations are:

10x + y + 10y + x = 66

and

x - y = 2

. Solving these, we find the number is 24.

Chapter Concept:

Formulating Linear Equations from Word Problems

The system has a unique solution.

The system has infinitely many solutions.

The system has no solution.

The system is inconsistent.

Correct Answer: B

Solution:

The equations are multiples of each other, indicating they are coincident lines and have infinitely many solutions.

Chapter Concept:

Algebraic Methods for Solving Equations

x = 4, y = 3

x = 2, y = 3

x = 3, y = 2

x = 5, y = 2

Correct Answer: A

Solution:

Substitute

x = 2y

from the second equation into the first to get

3(2y) + 4y = 20

, solving gives

y = 3

and

x = 6

Chapter Concept:

Solving Linear Equations

Fixed charge = ₹30, Additional charge per hour = ₹5

Fixed charge = ₹20, Additional charge per hour = ₹10

Fixed charge = ₹40, Additional charge per hour = ₹2

Fixed charge = ₹25, Additional charge per hour = ₹5

Correct Answer: A

Solution:

Let the fixed charge be ₹x and the additional charge per hour be ₹y. From the given conditions, we have the equations:

x + 3y = 50

and

x + 7y = 70

. Solving these, we subtract the first equation from the second to get

4y = 20

, so

y = 5

. Substituting back, we find

x = 30

. Thus, the fixed charge is ₹30 and the additional charge per hour is ₹5.

Chapter Concept:

Solving Linear Equations

Intersecting lines

Parallel lines

Coincident lines

None of the above

Correct Answer: C

Solution:

The lines are coincident because the second equation is a multiple of the first, indicating infinitely many solutions.

Chapter Concept:

Solving Linear Equations

They intersect at a single point.

They are parallel.

They coincide.

They do not intersect.

Correct Answer: C

Solution:

The second equation is a multiple of the first, indicating that the lines coincide, representing infinitely many solutions.

Chapter Concept:

Graphical Representation of Linear Equations

The lines intersect at a single point.

The lines are parallel.

The lines coincide.

The lines form a triangle with the x-axis.

Correct Answer: C

Solution:

The equations are multiples of each other, indicating that the lines coincide, representing infinitely many solutions.

Chapter Concept:

Graphical Representation of Linear Equations

2 rides and 4 games

4 rides and 2 games

3 rides and 3 games

5 rides and 1 game

Correct Answer: B

Solution:

Let the number of rides be

x

and games be

y

. We have

x = 2y

and

3x + 4y = 20

. Solving these, we find

x = 4

and

y = 2

Chapter Concept:

Solving Linear Equations

2x + 3y = 9

and

4x + 6y = 18

x + 2y = 4

and

2x + 4y = 8

3x + 4y = 20

and

x - 2y = 0

x + y = 5

and

2x + 2y = 10

Correct Answer: C

Solution:

The pair of equations

3x + 4y = 20

and

x - 2y = 0

are consistent and intersect at a point, hence they have a unique solution. The other pairs are either coincident or parallel.

Chapter Concept:

Algebraic Methods for Solving Equations

Correct Answer: A

Solution:

Let the number of ₹ 50 notes be

x

and the number of ₹ 100 notes be

y

. We have the equations:

x + y = 25

and

50x + 100y = 2000

. Solving these, we find

x = 10

and

y = 15

. Thus, Meena received 10 notes of ₹ 50.

Chapter Concept:

Formulating Linear Equations from Word Problems

Substitution method

Elimination method

Graphical method

None of the above

Correct Answer: D

Solution:

The equations have no solution as they are inconsistent. The elimination method shows this by resulting in a false statement.

Chapter Concept:

Solving Linear Equations

₹ 5

₹ 6

₹ 7

₹ 8

Correct Answer: B

Solution:

Let the fixed charge be

f

and the charge per km be

c

. We have the equations:

f + 10c = 105

and

f + 15c = 155

. Solving these, we find

f = 55

and

c = 6

. Thus, the charge per km is ₹ 6.

Chapter Concept:

Formulating Linear Equations from Word Problems

Find the value of one variable in terms of the other.

Add the equations together.

Graph the equations.

Multiply the equations by constants.

Correct Answer: A

Solution:

The first step in the substitution method is to express one variable in terms of the other using one of the equations.

Chapter Concept:

Algebraic Methods for Solving Equations

The equations are consistent with a unique solution.

The equations are inconsistent with no solution.

The equations are dependent with infinitely many solutions.

The equations are consistent with multiple solutions.

Correct Answer: A

Solution:

The lines intersect at one point, indicating a unique solution, hence the equations are consistent.

Chapter Concept:

Solving Linear Equations

The system has a unique solution.

The system has infinitely many solutions.

The system has no solution.

The system is dependent.

Correct Answer: C

Solution:

Parallel lines indicate that the system of equations is inconsistent and has no solution.

Chapter Concept:

Algebraic Methods for Solving Equations

Intersecting lines

Parallel lines

Coincident lines

None of the above

Correct Answer: B

Solution:

The lines are parallel, as they have the same slope but different intercepts, indicating no solution.

Chapter Concept:

Interpreting Solutions of Linear Equations

The pair of equations has a unique solution.

The pair of equations has infinitely many solutions.

The pair of equations has no solution.

The pair of equations is inconsistent.

Correct Answer: B

Solution:

The equations

3x + 4y = 20

and

6x + 8y = 40

are multiples of each other, indicating they represent the same line. Therefore, they have infinitely many solutions, making them dependent and consistent.

Chapter Concept:

Solving Linear Equations

Unique solution

No solution

Infinitely many solutions

Cannot be determined

Correct Answer: A

Solution:

The lines intersect at a single point, indicating a unique solution.

Chapter Concept:

Consistency and Solutions of Linear Equations

\frac{4}{5}

\frac{5}{6}

\frac{7}{9}

\frac{3}{4}

Correct Answer: A

Solution:

Let the fraction be

\frac{x}{y}

. We have the equations:

\frac{x+2}{y+2} = \frac{9}{11}

and

\frac{x+3}{y+3} = \frac{5}{6}

. Solving these, we find

x = 4

and

y = 5

. Thus, the original fraction is

\frac{4}{5}

Chapter Concept:

Formulating Linear Equations from Word Problems

(3, 1)

(6, 0)

(0, 2)

(1, 0)

Correct Answer: B

Solution:

Solving the equations

x + 3y = 6

and

2x - 3y = 12

simultaneously, we find the intersection point is

(6, 0)

. Thus, the solution is

(6, 0)

Chapter Concept:

Interpreting Solutions of Linear Equations

3x + 4y = 20

and

6x + 8y = 40

x + 2y = 4

and

2x + 4y = 8

5x - 8y + 1 = 0

and

15x - 24y + 3 = 0

2x + 3y = 9

and

4x + 6y = 18

Correct Answer: C

Solution:

The pair

5x - 8y + 1 = 0

and

15x - 24y + 3 = 0

are inconsistent because they represent parallel lines (same slope but different intercepts) and hence have no solution.

Chapter Concept:

Interpreting Solutions of Linear Equations

They have a unique solution.

They have no solution.

They have infinitely many solutions.

They are inconsistent.

Correct Answer: C

Solution:

Both equations are equivalent, hence they have infinitely many solutions.

Chapter Concept:

Formulating Linear Equations from Word Problems

Unique solution

No solution

Infinitely many solutions

Cannot be determined

Correct Answer: B

Solution:

The lines represented by the equations are parallel, hence the pair of equations has no solution.

Chapter Concept:

Consistency and Solutions of Linear Equations

Consistent with a unique solution

Inconsistent

Consistent with infinitely many solutions

Dependent and inconsistent

Correct Answer: C

Solution:

Both equations represent the same line, hence the system is consistent with infinitely many solutions.

Chapter Concept:

Consistency and Solutions of Linear Equations

They are consistent with a unique solution.

They are inconsistent.

They are dependent and consistent.

They have infinitely many solutions.

Correct Answer: B

Solution:

The equations are inconsistent because they lead to a false statement when one is subtracted from the other.

Chapter Concept:

Interpreting Solutions of Linear Equations

₹ 9

₹ 12

₹ 15

₹ 6

Correct Answer: A

Solution:

Let the fixed charge be

x

and the additional charge per day be

y

. We have the equations:

x + 4y = 27

and

x + 2y = 21

. Solving these, we find

x = 9

and

y = 6

. Therefore, the fixed charge is ₹ 9.

Chapter Concept:

Solving Linear Equations

The lines intersect at a point.

The lines coincide.

The lines are parallel.

The lines form a triangle.

Correct Answer: C

Solution:

If the lines are parallel, the pair of linear equations has no solution and is inconsistent.

Chapter Concept:

Graphical Representation of Linear Equations

Pair of Linear Equations in Two Variables

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter Notes

Solving Linear Equations

Graphical Representation of Linear Equations

Consistency and Solutions of Linear Equations

Formulating Linear Equations from Word Problems

Algebraic Methods for Solving Equations

Interpreting Solutions of Linear Equations

Examples

Verification

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes & Exam Tips

Solving Linear Equations

Graphical Representation of Linear Equations

Consistency and Solutions of Linear Equations

Formulating Linear Equations from Word Problems

Algebraic Methods for Solving Equations

Interpreting Solutions of Linear Equations

Multiple Choice Questions

Q1.Champa went to a 'Sale' to purchase some pants and skirts. The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased. How many pants did Champa buy?

Correct Answer: A

Q2.Which method is used to solve a pair of linear equations by eliminating one variable?

Correct Answer: B

Q3.Consider the pair of linear equations: 3x+4y=123x + 4y = 123x+4y=12 and 6x+8y=246x + 8y = 246x+8y=24. What is the nature of the solutions for this pair of equations?

Correct Answer: C

Q4.What does it mean when two linear equations are dependent?

Correct Answer: C

Q5.Given the pair of linear equations 5x−3y=115x - 3y = 115x−3y=11 and −10x+6y=−22-10x + 6y = -22−10x+6y=−22, determine the relationship between the lines.

Correct Answer: C

Q6.If the pair of linear equations 3x+4y=203x + 4y = 203x+4y=20 and x−2y=0x - 2y = 0x−2y=0 are graphed, what is the nature of their intersection?

Correct Answer: A

Q7.Consider the equations 5x−8y+1=05x - 8y + 1 = 05x−8y+1=0 and 3x−245y+35=03x - \frac{24}{5}y + \frac{3}{5} = 03x−524​y+53​=0. What is the nature of their solutions?

Correct Answer: C

Q8.A fraction becomes 911\frac{9}{11}119​ if 2 is added to both the numerator and the denominator. If 3 is added to both, it becomes 56\frac{5}{6}65​. What is the original fraction?

Correct Answer: A

Q9.A pair of linear equations is given by x+2y=4x + 2y = 4x+2y=4 and 2x+4y=122x + 4y = 122x+4y=12. What is the nature of the solution for this pair?

Correct Answer: C

Q10.Which method can be used to solve a pair of linear equations if the graphical method is not convenient?

Correct Answer: C

Q11.In the context of linear equations, what does it mean if the lines are parallel?

Correct Answer: B

Q12.If the lines represented by the equations 3x+2y=53x + 2y = 53x+2y=5 and 6x+4y=106x + 4y = 106x+4y=10 are plotted on a graph, what can be said about their relationship?

Correct Answer: C

Q13.Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Meena got 25 notes in all. How many ₹ 50 notes did she receive?

Correct Answer: B

Q14.If the lines represented by the equations 3x+4y=203x + 4y = 203x+4y=20 and x−2y=0x - 2y = 0x−2y=0 intersect, what can be said about the pair of equations?

Correct Answer: A

Q15.Consider the pair of linear equations: 3x+4y=203x + 4y = 203x+4y=20 and 2x−3y=72x - 3y = 72x−3y=7. Which method would be most efficient to find the solution, and what is the solution?

Correct Answer: C

Q16.If the lines represented by two linear equations are coincident, what can be said about the solutions?

Correct Answer: C

Q17.For the equations 2x+3y=72x + 3y = 72x+3y=7 and 4x+6y=144x + 6y = 144x+6y=14, determine the nature of the lines represented by these equations.

Correct Answer: C

Q18.If the pair of linear equations 4x−5y=204x - 5y = 204x−5y=20 and 8x−10y=408x - 10y = 408x−10y=40 are plotted on a graph, what will be the result?

Correct Answer: C

Q19.A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. What is the fixed charge?

Correct Answer: A

Q20.Consider the equations x−2y=0x - 2y = 0x−2y=0 and 3x+4y=203x + 4y = 203x+4y=20. What is the nature of the solution for this pair of linear equations?

Correct Answer: A

Q21.A shop sells pencils and erasers. The cost of 2 pencils and 3 erasers is ₹9, and the cost of 4 pencils and 6 erasers is ₹18. What can be concluded about the cost of pencils and erasers?

Correct Answer: C

Q22.If the equations 5x−8y+1=05x - 8y + 1 = 05x−8y+1=0 and 3x−245y+35=03x - \frac{24}{5}y + \frac{3}{5} = 03x−524​y+53​=0 are given, what transformation is needed to determine the nature of their solutions?

Correct Answer: B

Q23.Which of the following pairs of linear equations represents coincident lines?

Correct Answer: A

Q24.The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, what is the original number?

Correct Answer: A

Q25.If the equations 2x+3y=82x + 3y = 82x+3y=8 and 4x+6y=164x + 6y = 164x+6y=16 are given, what can be concluded about the system?

Correct Answer: B

Q26.What is the solution to the pair of equations 3x+4y=203x + 4y = 203x+4y=20 and x−2y=0x - 2y = 0x−2y=0 using the substitution method?

Correct Answer: A

Q27.A company offers a fixed charge for the first 5 hours of parking and an additional charge for each subsequent hour. If a customer pays ₹50 for parking for 8 hours and ₹70 for parking for 12 hours, what are the fixed charge and the additional charge per hour?

Correct Answer: A

Q28.If the lines represented by the equations x+2y=4x + 2y = 4x+2y=4 and 2x+4y=82x + 4y = 82x+4y=8 are plotted on a graph, what will be the nature of the lines?

Correct Answer: C

Q29.If the pair of linear equations x+2y=4x + 2y = 4x+2y=4 and 2x+4y=82x + 4y = 82x+4y=8 are graphed, what will be the nature of their intersection?