# Important Questions Class 10 Maths Chapter 3 Linear Equations in Two Variables

The mathematical equation which can be shown as â€˜ax by c = 0â€™ where a, b, and c are real numbers and both â€˜aâ€™ and â€˜bâ€™ are not â€˜0â€™ are called linear equations in two variables â€˜xâ€™ and â€˜yâ€™. The solution of such an equation is a pair of values, one value for â€˜xâ€™ and one value for â€˜yâ€™ which causes the equality of the equation on both sides (LHS = RHS). Such an equation can also be plotted geometrically over a sheet of paper or graph. in this article explore, Important Questions Class 10 Maths Chapter 3 Linear Equations in Two Variables.

Students looking for Â important questions in class 10 maths Chapter 3 Linear Equations in Two Variables should focus on the following topics:

- Consistency and inconsistency of linear equations
- Infinitely many solutions and unique solution of linear equations
- Algebraic methods of linear equation
- Substitution method
- Elimination method
- Cross multiplication method
- Geometrical/graphical solution of linear equations

## Important Questions Chapter 3 – Linear Equations in Two Variables

**Q1. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.**

**Solution:**

Given, half the perimeter of a rectangular garden = 36 m

so, 2(l + b)/2 = 36

(l + b) = 36 â€¦â€¦â€¦.(1)

Given, the length is 4 m more than its width.

Let width = x

And length = x + 4

Substituting this in eq(1), we get;

x + x + 4 = 36

2x + 4 = 36

2x = 32

x = 16

Therefore, the width is 16 m and the length is 16 + 4 = 20 m.

**Q2. Use elimination method to find all possible solutions of the following pair of linear equation:**

**2x + 3y = 8Â **

**4x + 6y = 7**

**Solution:**

Given,

2x + 3y = 8â€¦.(i)

4x + 6y = 7â€¦.(ii)

Multiply Equation (1) by 2 and Equation (2) by 1 to make the coefficients of x equal.

4x + 6y = 16â€¦.(iii)

4x + 6y = 7â€¦.(iv)

Subtracting (iv) from (iii),

4x + 6y â€“ 4x â€“ 6y = 16 â€“ 7

0 = 9, it is not possible

Therefore, the pair of equations has no solution.

**Q3. Five years ago, A was thrice as old as B and ten years later, A shall be twice as old as B. What are the present ages of A and B?**

**Solution:**

Let the present ages of B and A be x years and y years respectively. Then

Bâ€™s age 5 years ago = (x â€“ 5) years

and Aâ€™s age 5 years ago = (- 5) years

(-5) = 3 (x â€“ 5) = 3x â€“ y = 10 â€¦â€¦.(i)

Bâ€™s age 10 years hence = (x + 10) years

Aâ€™s age 10 years hence = (y + 10) years

y + 10 = 2 (x + 10) = 2x â€“ y = -10 â€¦â€¦.. (ii)

On subtracting (ii) from (i) we get x = 20

Putting x = 20 in (i) we get

(3 Ã— 20) â€“ y = 10 â‡’ y = 50

âˆ´ x = 20 and y = 50

Hence, Bâ€™s present age = 20 years and Aâ€™s present age = 50 years.

**Q4. Find the value(s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.**

**Solution:**

Given,

x + 2y = 5

3x + ky + 15 = 0

Also, given that the pair of equations has a unique solution.

Comparing the given equations with standard form,

a1 = 1, b1 = 2, c1 = -5

a2 = 3, b2 = k, c2 = 15

Condition for unique solution is:

a1/a2 â‰ b1/b2

1/3 â‰ 2/k

k â‰ (2)(3)

k â‰ 6

Thus, for all real values of k except 6, the given pair of equations has a unique solution.

**Q5. Solve the following pair of linear equations by the substitution and cross-multiplication methods:**

**8x + 5y = 9**

**3x + 2y = 4**

**Solution:**

8x + 5y = 9 â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(1)

3x + 2y = 4 â€¦â€¦â€¦â€¦â€¦â€¦.â€¦.(2)

From equation (2) we get;

x = (4 â€“ 2y) / 3 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3)

Substituting this value in equation 1, we get

8[(4 â€“ 2y)/3] + 5y = 9

32 â€“ 16y + 15y = 27

-y = -5

y = 5 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(4)

Substituting this value in equation (2), we get

3x + 10 = 4

3x = -6

x = -2

Thus, x = -2 and y = 5.

Now, Using Cross Multiplication method:

8x +5y â€“ 9 = 0

3x + 2y â€“ 4 = 0

x/(-20 + 18) = y/(-27 + 32 ) = 1/(16 â€“ 15)

-x/2 = y/5 = 1/1

âˆ´ x = -2 and y =5.

**Q6. The sum of the digits of a two digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number.**

**Solution:**

Let unit and tens digit be x and y.

âˆ´ Original number = 1x + 10y â€¦(i)

Reversed number = 10x + 1y

According to question,

x + y = 8

â‡’ y = 8 â€“ x â€¦(ii)

Also, 1x + 10Oy â€“ (10x + y) = 18

â‡’ x + 10y â€“ 10x â€“ y = 18

â‡’ 9y â€“ 9x = 18

â‡’ y â€“ x = 2 â€¦[Dividing both sides by 9

â‡’ 8 â€“ x â€“ x = 2 â€¦[From (it)

â‡’ 8 â€“ 2 = 2x

â‡’ 2x = 6

From (it), y = 8 â€“ 3 = 5

From (i), Original number = 3 + 10(5) = 53

**Q7. Solve the following pairs of equations by reducing them to a pair of linear equations:**

**1/2x + 1/3y = 2**

**1/3x + 1/2y = 13/6**

**Solution:Â **

Given,

1/2x + 1/3y = 2

1/3x + 1/2y = 13/6

Let us assume 1/x = m and 1/y = n , then the equations will change as follows.

m/2 + n/3 = 2

â‡’ 3m+2n-12 = 0â€¦.(1)

m/3 + n/2 = 13/6

â‡’ 2m+3n-13 = 0â€¦.(2)

Now, using cross-multiplication method, we get,

m/(-26-(-36) ) = n/(-24-(-39)) = 1/(9-4)

m/10 = n/15 = 1/5

m/10 = 1/5 and n/15 = 1/5

So, m = 2 and n = 3

1/x = 2 and 1/y = 3

x = 1/2 and y = 1/3

**Q8. The coach of a cricket team buys 7 bats and 6 balls for Rs.3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball.**

**Solution:**

Let the cost of a bat be x and the cost of a ball be y.

According to the question,

7x + 6y = 3800 â€¦â€¦â€¦â€¦â€¦â€¦. (i)

3x + 5y = 1750 â€¦â€¦â€¦â€¦â€¦â€¦. (ii)

From (i), we get;

y = (3800 â€“ 7x)/6 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (iii)

Substituting (iii) in (ii). we get,

3x + 5[(3800 â€“ 7x)/6] = 1750

â‡’3x + (9500/3) â€“ (35x/6) = 1750

3x â€“ (35x/6) = 1750 â€“ (9500/3)

(18x â€“ 35x)/6 = (5250 â€“ 9500)/3

â‡’-17x/6 = -4250/3

â‡’-17x = -8500

x = 500

Putting the value of x in (iii), we get;

y = (3800 â€“ 7 Ã— 500)/6 = 300/6 = 50

Hence, the cost of a bat is Rs 500 and the cost of a ball is Rs 50.

**Q9. Solve 2x + 3y = 11 and 2x â€“ 4y = â€“ 24 and hence find the value of â€˜mâ€™ for which y = mx + 3.**

**Solution:**

2x + 3y = 11â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(i)

2x â€“ 4y = -24â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (ii)

From equation (i), we get;

x = (11 â€“ 3y)/2 â€¦â€¦â€¦.â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(iii)

Putting the value of x in equation (ii), we get

2[(11 â€“ 3y)/2] â€“ 4y = âˆ’24

11 â€“ 3y â€“ 4y = -24

-7y = -35

y = 5â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(iv)

Putting the value of y in equation (iii), we get;

x = (11 â€“ 15)/2 = -4/2 = âˆ’2

Hence, x = -2, y = 5

Also,

y = mx + 3

5 = -2m +3

-2m = 2

m = -1

Therefore, the value of m is -1.

**Q10. On reversing the digits of a two digit number, number obtained is 9 less than three times the original number. If difference of these two numbers is 45, find the original number.**

**Solution:**

Let unitâ€™s place digit be x and tenâ€™s place digit bey.

âˆ´ Original number = x + 10y Reversed number = 10x + y

According to the Question,

10x + y = 3(x + 10y) â€“ 9

â‡’ 10x + y = 3x + 30y â€“ 9

â‡’ 10x + y â€“ 3x â€“ 30y = -9

â‡’ 7x â€“ 29y = -9 â€¦(i)

10x + y â€“ (x + 10y) = 45

â‡’ 9x â€“ 9y = 45

â‡’ x â€“ y = 5 â€¦[Dividing both sides by 9

â‡’ x â€“ 5 + y â€¦(ii)

Solving (i),

7x â€“ 29y = -9

7(5 + y) â€“ 29y = -9 â€¦[From (ii)

35+ 7y â€“ 29y = -9

-22y = -9 â€“ 35

-22y = -44 â‡’ y = 44/22 = 2

Putting the value of y in (ii),

x = 5 + 2 = 7

âˆ´ Original number = x + 10y

= 7 + 10(2) = 27

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