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
a_{1} x + b_{1} y + c_{1} = 0
a_{2} x + b_{2} y + c_{2} = 0
a) a_{1}/a_{2} ≠ b_{1}/b_{2} (Intersecting Lines/one Solution)
b) a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2} (Coincident Lines/Infinitely many Solutions)
c) a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2} (Parallel Lines/No solution)
(i) x + y = 5, 2x + 2 y = 10
a_{1}/a_{2}= 1/2
b_{1}/b_{2}= 1/2
c_{1}/c_{2}= -5/-10 = 1/2
From above
a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}
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 + 2 y - 10 = 0
2 y = 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
a_{1}/a_{2}= 1/3
b_{1}/b_{2}= -1/-3 = 1/3
c_{1}/c_{2}= -8/-16 = 1/2
From above
a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2}
Therefore, lines are parallel and have no solution.
Hence, the pair of equations are inconsistent.
x - y - 8 = 0
y = x - 8
x |
8 |
6 |
y = x - 8 |
0 |
-2 |
3x - 3y -16 = 0
3y = 3x -16
y = (3x -16)/3
x |
2 |
4 |
y = (3x -16)/3 |
-3.3 |
-1.3 |
(iii) 2x + y - 6 = 0, 4x - 2 y - 4 = 0
a_{1}/a_{2}= 2/4 = 1/2
b_{1}/b_{2}= 1/-2 = -1/2
c_{1}/c_{2}= -6/-4 = 3/2
From above
a_{1}/a_{2} ≠ b_{1}/b_{2}
Therefore, lines are intersecting and have one solution.
Hence, they are consistent.
2x + y - 6 = 0
y = 6 - 2x
x |
0 |
2 |
y = 6 - 2x |
6 |
2 |
4x - 2 y - 4 = 0
2 y = 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 - 2 y - 2 = 0, 4x - 4 y - 5 = 0
a_{1}/a_{2}= 2/4 = 1/2
b_{1}/b_{2}= -2/-4 = 1/2
c_{1}/c_{2}= -2/-5 = 2/5
From above
a_{1}/a_{2} = b_{1}/b_{2} ≠ c_{1}/c_{2}
Therefore, lines are parallel and have no solution.
Hence, the pair of equations are inconsistent.
2x - 2 y - 2 = 0
2 y = 2x - 2
y = x -1
x |
1 |
3 |
y = x -1 |
0 |
2 |
4x - 4 y - 5 = 0
4 y = 4x - 5
y = (4x - 5)/4
x |
4 |
3 |
y = (4x - 5)/4 |
2.8 |
1.8 |
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:
Which of the following pairs of linear equations are consistent / inconsistent? If consistent, obtain the Solution graphically: (i) x + y = 5, 2x + 2 y = 10 (ii) x - y = 8, 3x - 3y =16 (iii) 2x + y - 6 = 0, 4x - 2 y - 4 = 0 (iv) 2x - 2 y - 2 = 0, 4x - 4 y - 5 = 0
On comparing the ratios of the coefficients, we can say that the lines are parallel to each other and inconsistent in nature.