In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
3x-5y=206x-10y=40
Answers
infinitely many solutions
Step-by-step explanation:
Given:
3x -5y = 20
6x -10y = 40
So we get 3x -5y - 20 = 0
6x -10y - 40 = 0
a1 = 3, b1 = -5, c1 = -20
a2 = 6, b2 = -10, c2 = -40
a1/a2 = b1/b2 = c1/c2 = 1/2
Hence we can conclude that the given system of equation has infinitely many solutions.
Concept :
The general form for a pair of linear equations in two variables x and y is
a1x + b1y + c1 = 0 ,
a2x + b2y + c2 = 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.
Solution :
Given :
3x – 5y = 20 , 6x – 10y = 40
We can write these linear equations in the general form :
3x – 5y - 20 = 0
6x -10y -40 =0
On comparing with ax + by + c = 0 :
a1 = 3, b1 = -5, c1 = - 20
a2 = 6, b2 = 10, c2 = - 40
We have ,
a1/a2 = 3/6 = 1/2
b1/b2 = -5/-10 = 1/2
c1/c2 = -20/-40 = 1/2
Clearly
a1/a2 = b1/b2 = c1/c2
Therefore, the given sets of lines will be overlapping each other i.e., the lines will be coincident to each other and thus, there are infinitely many solutions .
Hope this answer will help you…
Some more questions from this chapter :
Which of the following pairs of linear equations has unique solution, no solution or infinitely many solutions? In case there is a
unique solution, find it by using cross multiplication method:
(i) x-3y-3=0 (ii) 2x+y=5
3x-9y-2=0 3x+2y=8
(iii) 3x-5y=20 (iv) x-3y-7=0
6x-10y=40 3x-3y-15=0
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In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
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