Question

$2x-5y=a$
$bx+10y=-8$
In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a ?

• A. $-4$
• B. $8$
• C. $4$
• D. $16$

Answer 1

The correct option is C $4$

The given equations are :

$2x+(-5)y+(-a)=0$ ...$\left(1\right)$

$bx+10y+8=0$ ...$\left(2\right)$

As we can see they are of the form $ax+by+c=0$

Also the condition for a system of equations to have infinitely many solutions is,

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

Here $a_1,a_2,b_1,b_2,c_1,c_2$ all are contants, and their values are respectively:

$a_1=2$

$a_2=b$

$b_1=-5$

$b_2=10$

$c_1=-a$

$c_2=8$

So if the given system of equations has infinitely many solutions then, $\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\Rightarrow \frac{2}{b}=\frac{-5}{10}=\frac{-a}{8}$

$\Rightarrow \frac{2}{b}=\frac{-1}{2}$

or $\Rightarrow \frac{-1}{2}=\frac{-a}{8}$

Hence $a=4$ and $b=-4$