Question

find the values of a & b, following system of linear equation has infinite number of solutions : 2x-3y=7 ; (a+b)x-(a+b-3)y = 4a+b.

Answer 1

Value of a, b = -5, -1 derived from the system of linear equation given which has infinite number of solution.

Solution:

For an equation to have infinite solution, the equations should be same as for now let us equate the equations and then solve it

2x - 3y=7  

(a+b)x-(a+b-3)y = 4a+b

Equate the value of ax + by = cn in both cases, after this we get:

a+b=2n

a+b-3=-3n and 4a+b=7n.  

Solving these we get the value of n = -3.

Putting the value of n, we get a+b=-6

a+b-3=18  

4a+b=-21

Solving the above equation, we get:

a=-6-b

-4(6+b)+b=-21

b= -1  

Putting the value of b in a = -6-b

a = -5.

Therefore, the value of a & b = -5, -1

Answer 2

- 5 and-1 are the value of a and b  

Given:

$2 x-3 y=7 ;(a+b) x-(a+b-3) y=4 a+b$

To find:

The values of a and b =?

Solution:

The two equations given are, $2 x-3 y=7 \text { and }(a+b) x-(a+b-3) y=4 a+b$

Comparing the two to get infinite value is  

$\frac{2}{a+b}=\frac{-3}{a+b-3}=\frac{7}{4 a+b}$

Solve the two equations one by one first separate

$\frac{2}{a+b}=\frac{-3}{a+b-3}$

$a+b=-6$

Now solving the second half we get  

$\begin{array}{l}{\frac{-3}{a+b-3}=\frac{7}{4 a+b}} \\ {-3(4 a+b)=7(a+b-3)} \\ {a=5 b}\end{array}$

Solving the two halves we get the value of a and b which is  

$\begin{array}{l}{a+b=-6, a=5 b} \\ {5 b+b=-6} \\ {b=-1}\end{array}$

Putting the value of b in $a+b=-6$, we get

$\begin{array}{l}{a-1=-6} \\ {a=-5}\end{array}$

Therefore, the value of “a” and “b” is equal to -5 and -1 respectively.