Systems of Equations Question 8

If the following equation has infinately many solutions for $x$ , what is the value of $b$ , given $a$ and $b$ are constants? $a(x+b)=5x-10$

-10

-2

2

5

Approach

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}