Math, asked by ThePROkillerYT, 10 months ago

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Answers

Answered by mahadevathani2003
0

Step-by-step explanation:

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Answered by AlluringNightingale
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Answer :

a = -1/3 , b = -5/3

Note:

★ A linear equation is two variables represent a straight line .

★ The word consistent is used for the system of equations which consists any solution .

★ The word inconsistent is used for the system of equations which doesn't consists any solution .

★ Solution of a system of equations :

It refers to the possibile values of the variable which satisfy all the equations in the given system .

★ A pair of linear equations are said to be consistent if their graph ( Straight line ) either intersect or coincide each other .

★ A pair of linear equations are said to be inconsistent if their graph ( Straight line ) are parallel .

★ If we consider equations of two straight line Ax + By + C = 0 and

A'x + B'y + C' = 0 , then ;

• The lines are intersecting if

A/A' ≠ B/B' .

→ In this case , unique solution is found .

• The lines are coincident if

A/A' = B/B' = C/C' .

→ In this case , infinitely many solutions are found .

• The lines are parallel if

A/A' = B/B' ≠ C/C' .

→ In this case , no solution is found .

Solution :

The given linear equations are ;

2x - 3y = 7

(a - b)x + (a + b)y = 3a + b - 2

The given equations can be rewritten as ;

2x - 3y - 7 = 0

(a - b)x + (a + b)y - (3a + b - 2) = 0

Clearly ,

A = 2

A' = a - b

B = -3

B' = a + b

C = -7

C' = -(3a + b - 2)

Thus ,

For the given linear equations to have infinitely many solutions , we have ;

A/A' = B/B' = C/C'

Considering A/A' = B/B' , we get ;

=> 2/(a - b) = -3/(a + b)

=> 2×(a + b) = -3×(a - b)

=> 2a + 2b = -3a + 3b

=> 2a + 3a = 3b - 2b

=> 5a = b

=> b = 5a -------------(1)

Now ,

Considering B/B' = C/C'

=> -3/(a + b) = -7/-(3a + b - 2)

=> -3/(a + b) = 7/(3a + b - 2)

=> 3×(3a + b - 2) = 7×(a + b)

=> 9a + 3b - 6 = 7a + 7b

=> 9a - 7a + 3b - 7b = 6

=> 2a - 4b = 6

=> 2(a - 2b) = 6

=> a - 2b = 6/2

=> a - 2b = 3

=> a - 2×5a = 3 {using eq-(1)}

=> a - 10a = 3

=> -9a = 3

=> a = 3/-9

=> a = -1/3

Now ,

Putting a = -1/3 in eq-(1) , we get ;

=> b = 5a

=> b = 5 × (-1/3)

=> b = -5/3

Hence ,

a = -1/3 , b = -5/3

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