Question 4 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
Class 10 - Math - Pair of Linear Equations in Two Variables Page 49
Answers
2) Inconsistent
3) Consistent
4) Inconsistent
The general form for a pair of linear equations in two variables x and y is
a1x + b1y + c1 = 0 ,
a2x + b2y + c2 = 0 ,
Where a1, a2, b1, b2, c1, c2 are all real numbers ,a1²+ b1² ≠ 0 & a2² + b2² ≠ 0.
Condition 1: Intersecting Lines
If a 1 / a 2 ≠ b 1 / b 2 , then the pair of linear equations has a unique solution.
Condition 2: Coincident Lines
If a 1 / a 2 = b 1 / b 2 = c 1 / c 2 ,then the pair of linear equations has infinite solutions.
A pair of linear equations, which has a unique or infinite solutions are said to be a consistent pair of linear equations.
A pair of linear equations, which has infinite many distinct common solutions are said to be a consistent pair or dependent pair of linear equations.
Condition 3: Parallel Lines
If a 1/ a 2 = b 1/ b 2 ≠ c 1 / c 2 , then a pair of linear equations has no solution.
A pair of linear equations which has no solution is said to be an inconsistent pair of linear equations.
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Solution:
(i) x + y = 5; x + y -5=0
2 x + 2 y = 10 2 x + 2 y - 10 =0
on comparing with ax+by+c+0
a1= 1 , b1=1, c1= -5
a2=2, b2=2, c2= - 10
a1/a2 = 1/2
b1/b2 = 1/2 &
c1/c2 = 5/10 = 1/2
Hence, a1/a2 = b1/b2 = c1/c2
Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.
Now we need to solve it graphically
[ graph is in the attachment]
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(ii) x – y = 8, x -y -8=0
3x – 3y = 16, 3 x-3 y-16=0
on comparing with ax+by+c+0
a1= 1 , b1= -1, c1= -8
a2=3, b2=-3, c2= - 16
a1/a2 = 1/3
b1/b2 = -1/-3 = 1/3
c1/c2 = 8/16 = 1/2
Hence, a1/a2 = b1/b2 ≠ c1/c2
Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.
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(iii) 2x + y – 6 = 0,
4x – 2y – 4 = 0
on comparing with ax+by+c+0
a1= 2 , b1= 1, c1= -6
a2=4, b2=-2, c2= -4
a1/a2 = 2/4 = 1/2
b1/b2 = -1/2 and
c1/c2 = -6/-4 = 3/2
Hence, a1/a2 ≠ b1/b2
Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.
Now we need to solve it graphically.
[ graph is in the attachment]
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(iv) 2x – 2y – 2 = 0,
4x – 4y – 5 = 0
on comparing with ax+by+c+0
a1= 2 , b1= -2, c1= -2
a2=4, b2=-4, c2= -5
a1/a2 = 2/4 = 1/2
b1/b2 = -2/-4 = 1/2
c1/c2 = 2/5
Hence, a1/a2 = b1/b2 ≠ c1/c2
Therefore, these linear equations are parallel to each other and thus, have no possible solution.
Hence, the pair of linear equations is inconsistent.
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Hope this will help you....
