To determine what value [tex]\( h \)[/tex] must take for the equation [tex]\( h(-x + 5) = -8x + 40 \)[/tex] to have one solution, we can follow these steps:
1. Examine the Structure of the Equation:
The given equation is:
[tex]\[
h(-x + 5) = -8x + 40
\][/tex]
2. Distribute [tex]\( h \)[/tex] on the left side of the equation:
[tex]\[
h(-x + 5) = h(-x) + h(5)
\][/tex]
Simplifying further:
[tex]\[
= -hx + 5h
\][/tex]
3. Set the distributed form equal to the right side of the equation:
[tex]\[
-hx + 5h = -8x + 40
\][/tex]
4. Equate Coefficients of Like Terms:
For the equation to have one solution, the coefficients of [tex]\( x \)[/tex] and the constant terms on both sides must match.
- Compare the coefficients of [tex]\( x \)[/tex]:
[tex]\[
-h = -8
\][/tex]
Solving for [tex]\( h \)[/tex]:
[tex]\[
h = 8
\][/tex]
- Check the constant terms:
[tex]\[
5h = 40
\][/tex]
Substituting [tex]\( h = 8 \)[/tex] into [tex]\( 5h \)[/tex]:
[tex]\[
5 \times 8 = 40
\][/tex]
This confirms the constant terms are also equal.
Hence, the equation will have one solution when [tex]\( h \)[/tex] equals 8 because you get the same number of [tex]\( x \)[/tex] terms on either side of the equation and the constant terms also correctly align.
