For what value of A and B the following system of equations have infinitely many solutions 2x a 4 y 2b 1 4x -( a 1 y 5b 1?

Text Solution

Solution : Given,<br> `2x−(a−4)y=2b+1`<br> `4x−(a−1)y=5b−1`<br> comparing with `a_1x+b_1x+c_1=0` and `a_2x+b_2x+c_2=0` `a_1​=2`<br> `b_1​=−(−a−4)`<br> and `c_1​=2b+1`<br> and ` a_2​=4`<br> `b_2​=−(a−1)`<br> and `c_2​=5b−1 `<br> For infinite number of solutions , `(a_1)/(​a_2)​​=(b_1)/(​b_2)​​=(c_1)/(​c_2)`<br>​​ `=2/4​=(−(a−4)​)/(−(a−1))=(2b+1​)/(5b−1 )`<br> Considering `(a−4)/(a−1)​=1/2`<br>​ `a=7`<br> And now for b consider <br>`(2b+1)/(5b-1)​=1/2`<br>​ `b=3`<br> Therefore,the given system of equation will have infinitely many solution if `a=7` and `b=3`

Determine a and b for which the following system of linear equations has infinite number of solutions:

Asked by Topperlearning User | 04 Jun, 2014, 01:23: PM

For infinite number of solution,

Therefore, we have:

Answered by | 04 Jun, 2014, 03:23: PM

a = 7 and b = 3

Explanation:

Considering 2x – (a – 4)y = 2b + 1

Comparing with a1x + b1y = c1

We get, a1 = 2, b1 = – (a – 4) and c1 = 2b + 1

Now, considering 4x – (a – 1)y = 5b – 1

Comparing with a2x + b2y = c2

We get, a2 = 4, b2 = – (a – 1), and c2 = 5b – 1

For infinite numbers of solution,

`a_1/a_2 = b_1/b_2 = c_1/c_2`

∴ `2/4 = (-(a - 4))/(-(a - 1)) = (2b + 1)/(5b  - 1)`

Considering `(a - 4)/(a - 1) = 1/2`

⇒ 2a – 8 = a – 1, ∴ a = 7

And now for 'b'

`(2b + 1)/(5b - 1) = 1/2`

⇒ 4b + 2 = 5b – 1

∴ b = 3

∴ The given system of equation will have infinitely many solution if a = 7 and b = 3.