B is the midpoint of modifying above A C with barand D is the midpoint of modifying above C E with bar. Solve for x, given BD = 3x + 5 and AE = 4x + 20.

The figure shows triangle A C E with a segment B D connecting the sides. point B is on segment A C. point D is on segment C E.

1 answer

From the given information, we can deduce that BD is the average of AC and CE. Hence, we have

BD = (AC + CE)/2

Plugging in the given expressions for BD, AC, and CE, we get:

3x + 5 = (AC + CE)/2

We also know that AE = AC + CE. Plugging in the given expression for AE, we get:

4x + 20 = AC + CE

Now, we have two equations:

3x + 5 = (AC + CE)/2
4x + 20 = AC + CE

To solve for x, we can subtract the first equation from the second equation:

(4x + 20) - (3x + 5) = (AC + CE) - ((AC + CE)/2)

x + 15 = (AC + CE)/2

Multiplying both sides by 2, we get:

2x + 30 = AC + CE

Now, we can substitute this expression into the first equation:

3x + 5 = (2x + 30)/2

Multiplying both sides by 2, we get:

6x + 10 = 2x + 30

Subtracting 2x and 10 from both sides, we get:

4x = 20

Dividing both sides by 4, we get:

x = 5

Therefore, x = 5.