Question

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:
x-2y=85x-10y=10

Answer 1

no solution.

Step-by-step explanation:

Given:  

x -2y = 8

5x -10y = 10

So we get x -2y - 8 = 0

5x -10y - 10 = 0

a1 = 1, b1 = -2, c1 = -8

a2 = 5, b2 = -10, c2 = -10

a1/a2 = b1/b2 but not equal to c1/c2

Hence we can conclude that the given system of equation has no solution.

Answer 2

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 :  

x - 2y = 8

5x - 10y = 10    

We can write these linear equations in the general form :

x - 2y - 8  = 0    

5x -10y - 10 = 0

on comparing with ax + by + c = 0  :

a1 = 1,  b1 = -2,  c1 = - 8

a2 = 5,    b2 = - 10,  c2 = - 10

We have ,  

a1/a2 = ⅕  

b1/b2 = -2/-10 = 1/5  

c1/c2 = -8/-10 = 2/5

Clearly , a1/a2 = b1/b2 ≠ c1/c2

So, the given sets of lines are parallel to each other.  

Therefore, they will not intersect each other and thus, there will no solution for these equations.

Hope this answer will help you…

 

Some more questions from this chapter :  

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:

x - 3y = 3

3x - 9y = 2

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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:

3x - 5y = 20

6x - 10y = 40

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