To find the value of [tex]\( h \)[/tex] that will ensure the equation:
[tex]\[ h(-2x + 2) = -8(x - 8) \][/tex]
has a single unique solution, we need to ensure that the coefficient of [tex]\( x \)[/tex] on both sides of the equation are equal.
1. Start by simplifying both sides of the equation.
Left side:
[tex]\[ h(-2x + 2) = h(-2x) + h(2) = -2hx + 2h \][/tex]
Right side:
[tex]\[ -8(x - 8) = -8x + 64 \][/tex]
2. For the equation to have a single unique solution, the coefficients of [tex]\( x \)[/tex] on both sides need to be the same.
Compare the coefficients of [tex]\( x \)[/tex]:
[tex]\[ -2h = -8 \][/tex]
3. Solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{-8}{-2} = 4 \][/tex]
We also ensure that the constant terms (terms without [tex]\( x \)[/tex]) do not affect the number of solutions. The equation is balanced once the [tex]\( x \)[/tex] coefficients are aligned.
Thus, the number of [tex]\( x \)[/tex]'s on either side of the equation is the same (specifically, one on each side).
So the completed sentence should read:
This equation will have one solution when [tex]\( h = 4 \)[/tex] because you get one solution when you have 1 number of [tex]\( x \)[/tex]'s on either side of the equation.
