[Work Included]
In ABC, centroid D is on median AM.
AD = x + 5 and DM = 2x – 1
Find AM.

Question

[Work Included] 
In ABC, centroid D is on median AM. 
AD = x + 5 and DM = 2x – 1
Find AM.

anonymous · Answer

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anonymous · Answer

It looks more like an orthocenter, but for the sake of it, let's say it is a centroid.

anonymous · Answer

$$ x + 5 = 2x - 1$$$$5 + x - 2x = 2x - 2x - 1$$$$5 - x = -1$$

anonymous · Answer

Subtract five from both sides. 
$$-x = -1 - 5$$$$-x = -6$$

anonymous · Answer

$$x = 6$$

anonymous · Answer

Now I can plug in the missing numbers.
$$6 + 5 = 2(6) - 1$$$$11 = 12 - 1$$$$= 11$$

kinggeorge · Answer

I think you're incorrect in assuming that $x+5$ must equal $2x-1$. The centroid of a triangle is located at a point 2/3 of the way along the median, and not 1/2 of the way along the median.

anonymous · Answer

In that case, I'll tweak a few things here.
$$x + 5 = 2(2x - 1)$$$$x + 5 = 4x - 2$$ Subtracting 4 from both sides.

anonymous · Answer

$$5 - 3x = -2$$$$-3x = -2 - 5$$$$-3x = -7$$

anonymous · Answer

$$x = \frac{ -7 }{ -3 }$$

kinggeorge · Answer

That looks good to me. You can simplify that to $\dfrac{7}{3}$ if you want.

kinggeorge · Answer

You're welcome.

anonymous · Answer

@KingGeorge 
The answer was really 11. Guess I should have trusted my gut, ha.

kinggeorge · Answer

Oops :( I guess that would be my fault.

kinggeorge · Answer

However, I think we just forgot the last step in the problem. We found $x=7/3$, but AM is equal to $(x+5)+(2x-1)$. If we plug in our value for $x$ into this, we get$$\frac{7}{3}+4+\frac{14}{3}=\frac{21}{3}+4=7+4=11$$