Question

I need help on this mathematical equation B is the Midpoint of AC.

Answer 1

`x=2`

Explanation:

B is the midpoint of AC so AB = BC.

AB + BC = AC

AB + AB = AC

2(AB) = AC

`2\left(3\left(3x-1\right)\right)=5\left(2x+2\right)`

Expand:

`2*3\left(3x-1\right)=5(2x+2)`

`6\left(3x-1\right)=5(2x+2)`

`6*3x-6*1=5*2x+5*2`

`18x-6=10x+10`

Add 6 to both sides:

`18x-6+6=10x+10+6`

`18x=10x+16`

Subtract 10x from both sides:

`18x-10x=10x+16-10x`

`8x=16`

Divide both sides by 8:

`(8x)/(8)=(16)/(8)`

`x=2`

Answer 2

`x=2`

Explanation:

First, we know that the entire segment adds up to 5(2x+2). In other words:

`AB+BC=5(2x+2)`

Now, since B is the midpoint of AC, by the definition of midpoint, AB and BC are equivalent. So:

`AB=BC`

Since we know that, let's substitute BC for AB in the first equation:

`AB+BC=5(2x+2)\\`

This is equivalent to:

`AB+AB=5(2x+2)`

Combine like terms:

`2AB=5(2x+2)`

And since we know that AB is 3(3x-1), substitute:

`2(3(3x-1))=5(2x+2)`

Now, solve for x.

Distribute the left side:

`6(3x-1)=5(2x+2)`

Distribute both sides:

`18x-6=10x+10`

Add 6 to both sides:

`18x=10x+16`

Subtract 10x from both sides:

`8x=16`

Divide both sides by 8:

`x=2`

So, the value of x is 2.

And we're done!

Notes:

You didn't particularly state what you need to find. If you want to length of AB or BC, it would be:

`AB=BC=3(3(2)-1)=15`

And the entire length AC is:

`AC=2(15)=30`