To find the length of [tex]\(\overline{AM}\)[/tex], given the expressions for [tex]\(AM\)[/tex] and [tex]\(\overline{AB}\)[/tex], we will use the fact that [tex]\(M\)[/tex] is the midpoint of [tex]\(\overline{AB}\)[/tex], meaning [tex]\(AM\)[/tex] is half the length of [tex]\(\overline{AB}\)[/tex]. Here are the steps to solve this:
1. Start with the expressions for [tex]\(AM\)[/tex] and [tex]\(AB\)[/tex]:
[tex]\[
AM = 3x + 3
\][/tex]
[tex]\[
AB = 8x - 6
\][/tex]
2. Use the property of the midpoint:
Since [tex]\(M\)[/tex] is the midpoint, [tex]\(AM\)[/tex] is half of [tex]\(\overline{AB}\)[/tex]:
[tex]\[
AM = \frac{1}{2} AB
\][/tex]
3. Set up the equation using the given expressions:
[tex]\[
3x + 3 = \frac{1}{2} (8x - 6)
\][/tex]
4. Simplify the equation:
Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
2(3x + 3) = 8x - 6
\][/tex]
[tex]\[
6x + 6 = 8x - 6
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Combine like terms:
[tex]\[
6 + 6 = 8x - 6x
\][/tex]
[tex]\[
12 = 2x
\][/tex]
[tex]\[
x = 6
\][/tex]
6. Substitute the value of [tex]\(x\)[/tex] back into the expression for [tex]\(AM\)[/tex]:
[tex]\[
AM = 3x + 3
\][/tex]
[tex]\[
AM = 3(6) + 3
\][/tex]
[tex]\[
AM = 18 + 3
\][/tex]
[tex]\[
AM = 21
\][/tex]
So, the length of [tex]\(\overline{AM}\)[/tex] is [tex]\(\boxed{21}\)[/tex] units.
