Question

complete the sentence based on the equation shown below.
$h(-x - 3) = -8x + 8$
answer attempt 1 out of 2
the same constant
different constants
constants can be the same or different
$h = \square$ because you get one solution when
either side of the equation and

Explanation:

Step1: Isolate linear term in argument

Let $t = -x - 3$, solve for $x$:
$x = -t - 3$

Step2: Substitute $x$ into right-hand side

Substitute $x=-t-3$ into $-8x+8$:

$$\begin{align*} -8(-t-3)+8&=8t+24+8\\ &=8t+32 \end{align*}$$

Step3: Identify linear function form

Since $h(t)=8t+32$, substitute back $t=-x-3$ to verify, and match to $h(-x-3)=-8x+8$. The coefficient of $x$ in the argument is $-1$, so solve for $h$:
From $h(-x-3)=8(-x-3)+32=-8x-24+32=-8x+8$, so $h(x)=8x+32$. The coefficient of the linear term (slope) is $8$.

Step4: Analyze solution condition

A linear equation $a_1x+b_1=a_2x+b_2$ has one solution when $a_1
eq a_2$ (different coefficients for $x$, which correspond to different constants for the linear terms when rearranged). Here, when we write the equation as $h(-x-3)+8x-8=0$, the linear terms have different coefficients, so we get one solution when the coefficients of the variable terms (from expanding $h(-x-3)$ and the other side) result in different constants for the $x$ term.

Answer:

$h = 8$ because you get one solution when either side of the equation has different constants for the linear variable terms.
Dropdown selection: different constants