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All questions of Linear Equations in Two Variables for Class 10 Exam

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If am ≠ bl, then the system of equations, ax + by = c, lx + my = n
  • a)
    has a unique solution
  • b)
    has no solution
  • c)
    has infinitely many solutions
  • d)
    may or may not have a solution
Correct answer is option 'A'. Can you explain this answer?

Vivek Rana answered
If am ≠ bl, then the equations ax+by=c, lx+my=n has a unique solution.
Given,
Pair of lines represented by the equations
ax + by = c
lx + my = n
For unique solution
For infinite solutions
For no solution
Given,
This can be transformed into
Therefore, If am ≠ bl, then the equations ax+by=c, lx+my=n has a unique solution.

The sum of the digits of a two-digit number is 9. If 27 is added to it, the digit of number get reversed. The number is
  • a)
    25
  • b)
    72
  • c)
    63
  • d)
    36
Correct answer is option 'D'. Can you explain this answer?

Avinash Patel answered
Lets,
First digit number = x
Second digit number = y
Number = (x+10y)
A/Q,
x + y = 9 ...................... (i)
A/Q,
(x+10y) = (10x+y) + 27
x + 10y = 10x + y +27
9x - 9y = 27
9(x - y) = 27
x - y = 27/9
x - y = 3 ......................... (ii)
Equation (i) and (ii) we get,
x = 3
Putting the value of x in eq.(i)
we get,
y = 6
Number = (10x +y)
= 10 x 3 + 6
= 30 + 6
= 36

The pair of linear equations x + y + 10 = 0 and x + y – 7 = 0 has:
  • a)
    One solution
  • b)
    Infinitely many solutions
  • c)
    No solutions
  • d)
    Two solutions
Correct answer is option 'C'. Can you explain this answer?

Gaurav Kumar answered
We have a1, a2 the coefficients of x2,b1 and b2 coefficients of x and c1 and c2 the constant terms.So,a1a2=b1b2c1c2which is a case of parallel lines which which never meet. So there are no solutions obtainable for these equations.

The pair of equations y = 0 and y = - 7 has
  • a)
    one solution
  • b)
    two solutions
  • c)
    infinitely many solutions
  • d)
    no solution
Correct answer is option 'D'. Can you explain this answer?

Gaurav Kumar answered
The equation are y=0 and y=-7
y=0 is on the x-axis and y=-7 is the line parallel to the x-axes at a distance 7 units from y=0
The line will be parallel
if we try to solve these equations we get 0=7 which is absurd.
So the equations are inconsistent.
Therefore there is no solution.

The pair of equations 3x + 4y = k, 9x + 12y = 6 has infinitely many solutions if –
  • a)
    k = 2
  • b)
    k = 6
  • c)
    k = 6
  • d)
    k = 3
Correct answer is option 'A'. Can you explain this answer?

Naina Sharma answered
An equation has infinitely many solutions when the lines are coincident.
The lines are coincident when 
So 3x + 4y = k, 9x + 12y = 6 are coincident when

Six years hence a man's age will be three times the age of his son and three years ago he was nine times as old as his son. The present age of the man is –
  • a)
    28 years
  • b)
    30 years
  • c)
    32 years
  • d)
    34 years
Correct answer is option 'B'. Can you explain this answer?

Dr Manju Sen answered
Let the present age of man is x and of son is y.
Six years hence,
Man’s age =x+6
Son’s age=y+6
Man’s age is 3 times son’s age
x+6=3(y+6)
x+6=3y+18
x=3y+12    …...1
Three years ago,
Man’s age =x-3
Son’s age=y-3
Man’s age was 9 times as of son
x-3=9(y-3)
x-3=9y-27
x=9y-24   ….2
From 1 and 2
3y+12=9y-24
6y=36
y=6
x=3*6+12=18+12=30 years

 One equation of a pair of dependent linear equations is -5x + 7y = 2, the second equation can be :
  • a)
    -10x + 14y + 4 = 0
  • b)
    -10x – 14x + 4 =
  • c)
    10x – 14y = -4
  • d)
    10x + 14y + 4 =0
Correct answer is option 'C'. Can you explain this answer?

Vikram Kapoor answered
If a  system of two linear equation is consistent system and has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.So we have which is satisfied by 10x – 14y = -4 only.

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is
  • a)
    inconsistent
  • b)
    Both
  • c)
    consistent
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Amit Sharma answered
a1 / a2 = b1 / b2 = c1 / c2
2/4 = 3/6 = 5/10
1/2 = 1/2 = 1/2
So, a1 / a2 = b1 / b2 = c1 / c2 
When these are equal then it is consistent.
Therefore option (C) is correct .

The pair of linear equations 2kx + 5y = 7, 6x – 5y = 11 has a unique solution if –
  • a)
    k ≠ -3
  • b)
    k = 3
  • c)
    k = 5
  • d)
    k = -5
Correct answer is option 'A'. Can you explain this answer?

Rohit Sharma answered
Given :
2 k x + 5 y – 7 = 0  ...( i )
6 x – 5 y – 1 = 0   ... ( ii )
Pair of linear equations has a unique solution.
We know for unique solution.
Comparing from ( i ) and ( ii ) we have
Put these values in formula.
Thus we get answer many values of k but leaving k ≠ -3.

The pair of linear equations 2x + 5y = k, kx + 15y = 18 has infinitely many solutions if –
  • a)
    k = 3
  • b)
    k = 6
  • c)
    k = 9
  • d)
    k = 18
Correct answer is option 'B'. Can you explain this answer?

Vivek Rana answered
An equation has infinitely many solutions when the lines are coincident.
The lines are coincident when 
So 2x + 5y = k, kx + 15y = 18 are coincident when

 For what value of ‘K’ will the system of equations: 3x + y = 1, (2K – 1) x + (K – 1) y = 2K + 1 have no solution
  • a)
    3
  • b)
    2
  • c)
    1
  • d)
    -2
Correct answer is option 'B'. Can you explain this answer?

Krishna Iyer answered
 is a case of parallel lines which never meet. So there are no solutions obtainable for these equations. So equations are inconsistent
3x + y = 1, (2K – 1) x + (K – 1) y = 2K + 1
b1=1,b2=k-1,c1=-1,c2=-2k-1

Can you explain the answer of this question below:

The pair of equations y = 0 and y = -7 has :

  • A:

    no solution

  • B:

    infinitely many solutions

  • C:

    one solution

  • D:

    two solutions

The answer is a.

Raghav Bansal answered
y=0 is x-axis… since every point has y=0. y=-7 is a line parallel to x-axis passing through x=0,y=-7. So the two lines are parallel to each other and are inconsistent which means that it has no solutions because it will never meet.

The number of solutions of the pair of linear equations x + 2y – 8 = 0 and 2x + 4y = 16 are:
  • a)
    None
  • b)
    Infinitely many
  • c)
    0
  • d)
    1
Correct answer is option 'B'. Can you explain this answer?

Pooja Shah answered
We have the equations x + 2y – 8 = 0 and 2x + 4y = 16 Where 
Here  which is the case of coincident lines . So there are infinitely many solutions.

A system of simultaneous linear equations has infinitely many solutions if two lines:
  • a)
    are coincident
  • b)
    intersect at two points
  • c)
    are parallel
  • d)
    intersect at one point
Correct answer is option 'A'. Can you explain this answer?

Damini kumar answered
Explanation:

Simultaneous linear equations are equations with two or more variables that are to be solved at the same time. These equations can be represented by lines, and the solutions represent the points where these lines intersect.

When two lines intersect at one point, there is only one solution to the system of equations. When two lines are parallel, there is no solution to the system of equations. However, when two lines are coincident, they overlap each other and have infinite solutions.

Example:

Consider the system of equations:

2x + 3y = 6
4x + 6y = 12

We can solve this system of equations by using elimination or substitution method.

Using the elimination method, we can multiply the first equation by 2 and subtract the second equation from it to eliminate x, which gives:

4x + 6y = 12
- (4x + 6y = 12)
-----------------
0x + 0y = 0

This equation is always true, which means that the two equations are equivalent. Therefore, they represent the same line, and there are infinitely many solutions to this system of equations.

Using the substitution method, we can solve for y in the first equation and substitute it into the second equation, which gives:

y = (6 - 2x)/3
4x + 6((6 - 2x)/3) = 12

Simplifying the second equation, we get:

4x + 4x = 12

Which gives:

x = 3/2

Substituting this value of x into the first equation, we get:

2(3/2) + 3y = 6

Simplifying, we get:

3y = 3

Which gives:

y = 1

Therefore, the solution to this system of equations is (3/2, 1). However, this is just one solution, and there are infinitely many solutions to this system of equations since the two lines are coincident.

Which of the following pairs of equations represent inconsistent system?​
  • a)
    3x – y = -8 3x – y = 24
  • b)
    5x – y = 10 10x – 2y = 20
  • c)
    3x – 2y = 8 2x + 3y = 1
  • d)
    lx – y = m x + my = l
Correct answer is option 'A'. Can you explain this answer?

Amit Sharma answered
is a case of parallel lines which never meet. So there are no solutions obtainable for these equations. So equations are inconsistent.
3x – y = -8 ,3x – y = 24
3x – y +8=0 ,3x – y -24=0

So, Therefore the equations are inconsistent.

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)
Correct answer is option 'D'. Can you explain this answer?

Raghav Bansal answered
By graphically in every condition, if a, b>>0; a, b< 0, a>0, b< 0; a<0, b>0 but a = b≠ 0.
The pair of equations x = a and y = b graphically represents lines which are intersecting at (a, b).
If a, b > 0 

Similarly, in all cases two lines intersect at (a, b).

Which of the following equation is not a linear equation?​
  • a)
    2a-b =1
  • b)
    2a+b =1
  • c)
    a+b =1
  • d)
    √a+b =1
Correct answer is option 'D'. Can you explain this answer?

Gaurav Kumar answered
Linear equation is an equation between two variables that gives a straight line when plotted on a graph. Linear equations have degree 1 only which means that power of the variables is 1 only. Since does not have degree 1 its not a linear equation.

The pair of linear equations 3x + 7y = k, 12x + 2ky = 4k + 1 do not have any solution if
  • a)
    k = 7
  • b)
    k = 14
  • c)
    k = 21
  • d)
    k = 28
Correct answer is option 'B'. Can you explain this answer?

Anirudh Mukherjee answered
Explanation:

Given pair of linear equations are:

3x - 7y = k ...(1)

12x - 2ky = 4k - 1 ...(2)

We need to find the value of k for which these equations do not have any solution.

Method:

If a pair of linear equations do not have any solution, it means they are inconsistent. Two linear equations are inconsistent if their slopes are equal but their y-intercepts are not equal.

We can rearrange the given equations into slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.

Equation (1) can be rearranged as:

y = (3/7)x - (k/7) ...(3)

Equation (2) can be rearranged as:

y = (6/k)x - (2k-1)/(2k) ...(4)

Comparing equations (3) and (4), we get:

m1 = 3/7 and m2 = 6/k

For the pair of equations to be inconsistent, m1 = m2 and b1 ≠ b2.

Equating the slopes, we get:

3/7 = 6/k

Solving for k, we get:

k = 14

Now, substituting k = 14 in equations (1) and (2), we get:

3x - 7y = 14 ...(5)

12x - 28y = 55 ...(6)

Multiplying equation (5) by 4, we get:

12x - 28y = 56 ...(7)

Comparing equations (6) and (7), we see that the y-intercepts are not equal. Hence, the given pair of equations are consistent for k = 14.

Conclusion:

Thus, the given pair of linear equations 3x - 7y = k, 12x - 2ky = 4k - 1 do not have any solution if k = 14. Therefore, the correct option is (B).

The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have
  • a)
    Unique solution
  • b)
    Exactly two solutions
  • c)
    No solution
  • d)
    Infinitely many solutions
Correct answer is option 'C'. Can you explain this answer?

Meera Rana answered
Given, 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 
a1/a2 = 9/18 = 1/2 
b1/b2 = 3/6 = 1/2 
c1/c2 = 12/26 = 6/13 
Since, a1/a2 = b1/b2 ≠ c1/c2 
So, the pairs of equations are parallel and the lines never intersect each other at any point, therefore there is no possible solution.

8 girls and 12 boys can finish work in 10 days while 6 girls and 8 boys can finish it in 14 days. Find the time taken by the one girl alone that by one boy alone to finish the work.       
  • a)
    120, 130       
  • b)
    140,280       
  • c)
    240,280       
  • d)
    100,120
Correct answer is option 'B'. Can you explain this answer?

Vikas Kumar answered
Work done by 1 girl and 1 boy in x and y days respectively.
work done by 1 girl and 1 boy in 1 day is (1/x) and (1/y).
so, work done by 8 girls and 12 boys in 1 day is (8/x) + (12/y) = 1/10
let (1/x) = a and (1/y) = b
so, 8a + 12b = 1/10
→ 80a + 120b = 1 ---- (1)
work done by 6 girls and 8 boys in 1 day is (6/x) + (8/y) = 1/14
6a + 80 = 1/14
→ 84a + 112b = 1 ---- (2)
By elimination method, Multiple equation 1 by 21 on both sides, we get
1680a + 2520b = 21 ---- (3)
Multiply equation 2 by 20 on both sides, we get
1680a + 2240b = 20 ---- (4)
On solving equation 3 and 4, we get
2520 b - 2240b = 21 - 20
→ 280 b = 1
→ b = 1/280
b =1/y
→ 1/280 = 1/y
→ y = 280
80a + 120 x (1/280) = 1 (From 1)
→ 80a + (3/7) = 1
→ 80a = 1 - (3/7)
→ 80a = (7 - 3)/7
→ 80a = 4/7
→ a = 4/(7 × 80)
→ a = 1/140
→ a = 1/x
→ 1/140 = 1/x
→ x = 140
1 girl and 1 boy alone take 140 days and 280 days to complete a work.

If the pair of equation has no solution, then the pair of equation is :
  • a)
    inconsistent
  • b)
    coincident
  • c)
    consistent
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?

Vikas Kumar answered
If two lines are parallel then, they have no solution pair of linear equations is inconsistent;
If two lines are coincident then, they have infinite solution and pair of linear equations is consistent;
If two lines are intersecting then, they have unique solution and pair of linear equations is consistent.

Three chairs and two tables cost Rs. 1850. Five chairs and three tables cost Rs. 2850. Then the total cost of one chair and one table is –
  • a)
    Rs. 800
  • b)
    Rs. 850
  • c)
    Rs. 900
  • d)
    Rs. 950
Correct answer is option 'B'. Can you explain this answer?

Niyati nayar answered
Understanding the Problem
We have two equations based on the information provided about chairs and tables:
1. Equation from the first statement:
- 3 chairs + 2 tables = Rs. 1850
- Let the cost of one chair be C and one table be T.
- This gives us the equation: 3C + 2T = 1850
2. Equation from the second statement:
- 5 chairs + 3 tables = Rs. 2850
- This gives us the equation: 5C + 3T = 2850
Solving the Equations
To find the values of C and T, we can use the method of elimination or substitution. Here, we will use elimination:
- From the first equation (3C + 2T = 1850), multiply everything by 3:
- 9C + 6T = 5550
- From the second equation (5C + 3T = 2850), multiply everything by 2:
- 10C + 6T = 5700
Now, we have two new equations:
1. 9C + 6T = 5550
2. 10C + 6T = 5700
Next, subtract the first equation from the second:
10C + 6T - (9C + 6T) = 5700 - 5550
C = 150
Now substitute C back into one of the original equations to find T:
3C + 2T = 1850
3(150) + 2T = 1850
450 + 2T = 1850
2T = 1400
T = 700
Calculating Total Cost
Now that we have the costs:
- Cost of one chair (C) = Rs. 150
- Cost of one table (T) = Rs. 700
The total cost of one chair and one table is:
C + T = 150 + 700 = Rs. 850
Conclusion
Thus, the total cost of one chair and one table is Rs. 850, which corresponds to option 'B'.

Solve : 5x/2+3x/4>39/4, x ∈ R :
  • a)
    (-3, ∞)
  • b)
    (3,3)
  • c)
    (-∞,3)
  • d)
    (3,∞)
Correct answer is option 'D'. Can you explain this answer?

Aishwarya jain answered
It is not clear what the expression is asking. Please provide more information or clarify the expression.

The pairs of equations x + 2y - 5 = 0 and -4x - 8y + 20 = 0 have:
  • a)
    Unique solution
  • b)
    Exactly two solutions
  • c)
    Infinitely many solutions
  • d)
    No solution
Correct answer is option 'C'. Can you explain this answer?

Meera Rana answered
a1/a2 = 1/-4
b1/b2 = 2/-8 = 1/-4
c1/c2 = -5/20 = -¼
This shows:
a1/a2 = b1/b2 = c1/c2
Therefore, the pair of equations has infinitely many solutions.

Solve : 2(x-1)/5<3(2+x)/7, x ∈ R :
  • a)
    (44, ∞)
  • b)
    (44,-∞)
  • c)
    (– 44,∞)
  • d)
    (+ 44,∞)
Correct answer is option 'C'. Can you explain this answer?

Drishti Kumari answered
2( x - 1 ) / 5 is less than 3 ( 2 + x ) / 7
2x - 2 / 5 is less than 6 + 3x / 7

Cross multiplication
14x - 14 is less than 30 + 15x
14 x - 15 x is less than 30 + 14
- x is less than 44
When we divide from negative no. to positive no. then inequality sign becomes change
So,
x is greater than - 44
On number line , it is ( - 44 , + infinity )
Hence, option (C) is correct .

Solve : x/5R : <(3x-2)/4-(5x-3)/5, x ∈ R:
  • a)
    (2/7,∞)
  • b)
    (2/9,∞)
  • c)
    (-∞, 2/9)
  • d)
    (∞, 2/9)
Correct answer is option 'C'. Can you explain this answer?

Ram Mohith answered
I think the question is to solve for x, x/5 < (3x - 2)/4 - (5x - 3)/5
Now multiply throughout with 20,
4x < 5(3x - 2) - 4(5x - 3)
4x < 15x - 10 - 20x + 12
20x - 15x + 4x < 12 - 10
9x < 2
x < 2/9
So, x can be any real number in the interval (-infinity, 2/9) 

In Fig., ABCD is a rectangle. Find the values of x and y. 
  • a)
    22, 8 
  • b)
    20,10
  • c)
    15, 9
  • d)
    -22, -8
Correct answer is option 'A'. Can you explain this answer?

Kds Coaching answered
Since ABCD is a rectangle
⇒ AB = CD and BC = AD
x + y = 30 …………….. (i)
x – y = 14 ……………. (ii)
(i) + (ii) ⇒ 2x = 44
⇒ x = 22
Plug in x = 22 in (i)
⇒ 22 + y = 30
⇒ y = 8

The solution of the equations x - y = 2 and x + y = 4 is:
  • a)
    3 and 1
  • b)
    4 and 3
  • c)
    5 and 1
  • d)
    -1 and -3
Correct answer is option 'A'. Can you explain this answer?

Kds Coaching answered
To solve the equations, follow these steps:
  • Start with the first equation: x - y = 2.
  • Rearrange to express x in terms of y: x = y + 2.
  • Substitute this expression for x into the second equation: x * y = 4.
  • This gives: (y + 2) * y = 4.
  • Expanding this results in: y^2 + 2y - 4 = 0.
  • Now, use the quadratic formula to solve for y: y = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = 2, and c = -4.
  • Calculating the discriminant: b² - 4ac = 2² - 4(1)(-4) = 4 + 16 = 20.
  • Thus, y = (-2 ± √20) / 2.
  • Calculating the two possible values for y gives: y = 1 or y = -4.
  • Substituting y = 1 back into x = y + 2 yields: x = 3.
  • For y = -4, substituting gives: x = -2.
Therefore, the solutions are:
  • x = 3 and y = 1.
  • x = -2 and y = -4.

Chapter doubts & questions for Linear Equations in Two Variables - Mathematics Class 10 (Maharashtra SSC Board) 2025 is part of Class 10 exam preparation. The chapters have been prepared according to the Class 10 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Class 10 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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