If the following equation has infinately many solutions for \(x\), what is the value of \(b\), given \(a\) and \(b\) are constants?
$$ a(x+b)=5x-10 $$
Although there is only one equation, this question corresponds to the following system:
$$ y=a(x+b)$$
$$ y=5x-10 $$
If a system has infinately many solutions, the equations should be identical.
Distribute \(a\) in the first equation and compare corresponding parts,
$$ y=\colorbox{yellow}{\(ax\)}+\colorbox{aqua}{\(ab\)}$$
$$ y=\colorbox{yellow}{5x}\colorbox{aqua}{-10} $$
Setting the linear portion equal to each other.
$$ ax=5x$$
$$a=5$$
Setting the constant portion equal to each other, and substituting the value of \(a\).
$$ ab=-10$$
$$(5)b=-10$$
$$b=\boxed{-2}$$