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:
2x - (a - 4)y = 2b + 1; 4x - (a - 1) y = 5b - 1.
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.