Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) x + y = 5, 2x + 2y = 10
(ii) x - y = 8, 3x - 3y =16
(iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0
(iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0
Solution:
For any pair of linear equation,
a₁ x + b₁ y + c₁ = 0
a₂ x + b₂ y + c₂ = 0
a) a₁/a₂ ≠ b₁/b₂ (Intersecting Lines/uniqueSolution)
b) a₁/a₂ = b₁/b₂ = c₁/c₂ (Coincident Lines/Infinitely many Solutions)
c) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Parallel Lines/No solution)
(i) x + y = 5, 2x + 2y = 10
a₁/a₂= 1/2
b₁/b₂= 1/2
c₁/c₂= -5/(-10) = 1/2
From the above,
a₁/a₂ = b₁/b₂ = c₁/c₂
Therefore, lines are coincident and have infinitely many solutions. Hence, they are consistent.
x + y - 5 = 0
y = - x + 5
y = 5 - x
x |
1 |
2 |
y = 5 - x |
4 |
3 |
2x + 2y - 10 = 0
2y = 10 - 2x
y = 5 - x
x |
3 |
4 |
y = 5- x |
2 |
1 |
All the points on coincident line are solutions for the given pair of equations.
(ii) x - y = 8, 3x - 3y =16
a₁/a₂ = 1/3
b₁/b₂ = -1/(-3) = 1/3
c₁/c₂ = - 8/(-16) = 1/2
From the above,
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Therefore, lines are parallel and have no solution.
Hence, the pair of equations are inconsistent.
(iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 1/(-2) = -1/2
c₁/c₂ = -6/(-4) = 3/2
From the above,
a₁/a₂ ≠ b₁/b₂
Therefore, lines are intersecting and have a unique solution.
Hence, they are consistent.
2x + y - 6 = 0
y = 6 - 2x
x |
0 |
2 |
y = 6 - 2x |
6 |
2 |
4x - 2y - 4 = 0
2y = 4x - 4
y = 2x - 2
x |
2 |
3 |
y = 2x - 2 |
2 |
4 |
x = 2 and y = 2 are solutions for the given pair of equations.
(iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = -2/(-4) = 1/2
c₁/c₂ = -2/(-5) = 2/5
From the above,
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Therefore, lines are parallel and have no solution.
Hence, the pair of equations are inconsistent.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 3
Video Solution:
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10 (ii) x - y = 8, 3x - 3y =16 (iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0 (iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0
NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.2 Question 4
Summary:
On comparing the ratios of the coefficients of the following pairs of linear equations, we see that (i) x + y = 5, 2x + 2y = 10 have infinitely many solutions. Hence, they are consistent. (ii) x - y = 8, 3x - 3y =16 are parallel and have no solution.Hence, the pair of equations are inconsistent. (iii) 2x + y - 6 = 0, 4x - 2y - 4 = 0 are intersecting and have a unique solution. Hence, they are consistent. (iv) 2x - 2y - 2 = 0, 4x - 4y - 5 = 0 are parallel and have no solution. Hence, the pair of equations are inconsistent.
☛ Related Questions:
- Form the pair of linear equations in the following problems and find their solutions graphically. (i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz. (ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one pencil and that of one pen.
- On comparing the ratios a1/a2 = b1/b2 = 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 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 18x + 6y + 24 = 0 (iii) 6x – 3y + 10 = 0 2x – y + 9 = 0
- On comparing the ratios find out whether the following pair of linear a1/a2,b1/b2 and c1/c2, find out whether the following pair of linear equations are consistent, or inconsistent (i) 3x + 2 y = 5; 2x - 3y = 7 (ii) 2x - 3y = 8; 4x - 6 y = 9 (iii) 3/2x + 5/3y = 7; 9x -10y = 14 (iv) 5x - 3y = 11; -10x + 6 y = -22 (v) 4/3x + 2 y = 8; 2x + 3y = 12