Class 10 - Chapter 3: Pair of Linear Equations in Two Variables

Important Concept

Source- NCERT

Exercise 3.1

Question-1: Form the pair of linear equations in the following problem, and find their solutions graphically.

Solution

Let the number of boys be x and the number of girls be y.

According to the question,

Total number of students = 10

Therefore,

x + y = 10

This is the first linear equation.

The number of girls is 4 more than the number of boys.

Therefore,

y = x + 4

Bringing all terms to one side,

y - x = 4

This is the second linear equation.

Hence, the required pair of linear equations is:

x + y = 10

y - x = 4

For Graphical Representation

We now find some points for each equation to draw the graph.

Equation 1: x + y = 10

Equation 2: y - x = 4

Graphical Solution

Plot the points of both equations on a graph paper and draw the two straight lines.

The two lines intersect at the point (3, 7).

Therefore,

x = 3

y = 7

Number of boys = 3

Number of girls = 7

Jab bhi graph paper par 2 lines intersect karengi
to us intersection point ke coordinates dekhne hain. wohi coordinates answers honge.
like is question me coordinates (3, 7) hain iska matlab ye hai ki is ordered pair me pehli value
x ki or dusri value y ki hogi. matlab x = 3 and y = 7. bas bat khatm

Q2. On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i) 5x − 4y + 8 = 0 and 7x + 6y − 9 = 0

a₁/a₂ = 5/7, b₁/b₂ = -4/6 = -2/3

Since a₁/a₂ ≠ b₁/b₂, the lines intersect at one point.

Answer: Intersecting lines

(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0

a₁/a₂ = 9/18 = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = 12/24 = 1/2

a₁/a₂ = b₁/b₂ = c₁/c₂

Answer: Coincident lines

(iii) 6x − 3y + 10 = 0 and 2x − y + 9 = 0

a₁/a₂ = 6/2 = 3

b₁/b₂ = -3/-1 = 3

c₁/c₂ = 10/9

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Answer: Parallel lines

Q3. On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5 ; 2x − 3y = 7

3/2 ≠ 2/-3

Consistent (Unique Solution)

(ii) 2x − 3y = 8 ; 4x − 6y = 9

2/4 = -3/-6 = 1/2

8/9 ≠ 1/2

Inconsistent

(iii) (3/2)x + (5/3)y = 7 ; 9x − 10y = 14

After simplifying: we get

9x + 10y = 42

9x − 10y = 14

Ratios are not equal.

Consistent (Unique Solution)

(iv) 5x − 3y = 11 ; -10x + 6y = -22

5/(-10) = -3/6 = 11/(-22)

All ratios are equal.

Consistent (Infinitely Many Solutions)

(v) (4/3)x + 2y = 8 ; 2x + 3y = 12

Multiplying first equation by 3:

4x + 6y = 24

Second equation × 2:

4x + 6y = 24

Consistent (Infinitely Many Solutions)

Q4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i) x + y = 5 and 2x + 2y = 10

Equation 1: x + y = 5

Two points are:

Equation 2: 2x + 2y = 10

Dividing by 2:

x + y = 5

Two points are:

Observation: Both equations represent the same line.

Graph Hence The two lines coincide.

Solution: Infinitely many solutions.

(ii) x - y = 8 and 3x - 3y = 16

Equation 1: x - y = 8

Two points are:

Equation 2: 3x - 3y = 16

Dividing by 3:

x - y = 16/3

Two points are:

Observation: The slopes are equal but constants are different.

Graph Hence The lines are parallel.

Solution: No solution.

(iii) 2x + y - 6 = 0 and 4x - 2y - 4 = 0

Equation 1:

2x + y = 6

y = 6 - 2x

Two points are:

Equation 2:

4x - 2y = 4

2x - y = 2

y = 2x - 2

Two points are:

Find the intersection point:

2x + y = 6

2x - y = 2

Adding both equations:

4x = 8

x = 2

Substituting in 2x + y = 6:

4 + y = 6

y = 2

Graph Hence The lines intersect at (2,2).

Solution: (2,2)

(iv) 2x - 2y - 2 = 0 and 4x - 4y - 5 = 0

Equation 1:

x - y = 1

Two points are:

Equation 2:

4x - 4y = 5

x - y = 5/4

Two points are:

Observation: The slopes are equal but constants are different.

Graph Hence The lines are parallel.

Solution: No solution.

Final Answers

Q5. 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.

Let width = x m

Length = (x + 4) m

Half perimeter = 36 m

x + (x + 4) = 36

2x + 4 = 36

2x = 32

x = 16

Length = 20 m

Answer:

• Width = 16 m
• Length = 20 m

Q6. Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines (ii) parallel lines (iii) coincident lines

Given equation:

2x + 3y − 8 = 0

(i) Intersecting Lines

Choose:

x + y − 4 = 0

Answer: x + y − 4 = 0

(ii) Parallel Lines

Choose:

4x + 6y − 10 = 0

a₁/a₂ = b₁/b₂ but c₁/c₂ different.

Answer: 4x + 6y − 10 = 0

(iii) Coincident Lines

Multiply by 2:

4x + 6y − 16 = 0

Answer: 4x + 6y − 16 = 0

(Note- For drawing graphs you can use the tool. click on the link below to draw the graph using desmos: )

Desmos- A graphing Calculator