Which of the following pairs of linear equations are consistent/inconsistent? If
consistent, obtain the solution graphically: x-y=8 3x-3y=16
Answers
Step-by-step explanation:
Graphical Method of solving pair of linear equations in two variables
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.
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
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.
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.