Question

Determine the values of a and b for which the following system of linear equation has infinite solutions.
$2x-(a-4)y=2b+1$
$4x-(a-1)y=5b-1$

• A. $a=3,b=7$
• B. $a=7,b=7$
• C. $a=7,b=3$
• D. $a=3,b=3$

Answer 1

Answer: (C)

$a=7,b=3$

$2x-(a-4)y=2b+1$

$4x-(a-1)y=5b-1$

Considering $2x-(a-4)y=2b+1$

Comparing with

$a_{1}x+b_{1}y=c_1{a}_1=2$

$b_1=-(-a-4)$

and $c_1=2b+1$

Now considering $4x-(a-1)y=5b-1$

Comparing with

$a_{2}x+b_{2}y=c_2$

$a_2=4$

$b_2=-(a-1)$

and $c_2=5b-1$

For infinite number of solutions ,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\therefore \frac{2}{4}=\frac{-(a-4)}{-(a-1)}=\frac{2b+1}{5b-1}$

Considering $\frac{a-4}{a-1}=\frac{1}{2}$

$\Rightarrow 2a-8=a-1\Rightarrow a=7$

And now for b consider $\frac{2b+1}{5b-1}=\frac{1}{2}$

$\Rightarrow 4b+2=5b-1$

$\therefore b=3$

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