Question

B is the midpoint of ac. Ab = x+9 and bc = 3x-7 find x and ac

Answer 1

To solve this problem, we need to know 2 relationships:

1. AC = AB + BC

The distance of AC is the sum of AB and BC.

$AC = AB + BC$

We know this since the distance of going from A to C (AC) is the same as going from A to B (AB), then B to C (BC).

2. AB = BC

The distance of AB is the same as AC.

$AB = BC$

We know this since B is in the middle of AC, so the distance from B to A (BA) is the same as the distance from B to C (BC).

You can see the attached image (at the bottom) for a visualization of this.

Putting them together

Since we know the values of AB and BC...

$AB = x+9\\BC = 3x-7$

...we can put these values into our 2nd equation and solve for x:

$AB = BC\\x + 9 = 3x -7$

Add 7 to both sides:

$x + 16 = 3x$

Subtract x from both sides:

$16 = 2x$

Divide both sides by 2:

$8 = x\\x = 8$

Knowing x, we can find the distance of AC using our first equation.

$AC = AB + BC$

Let's put in the values of AB and BC:

$AC = (x+9) + (3x-7)$

Before we put in x = 8, we can simplify this:

$AC = (x+9) + (3x-7)\\AC = x + 9 + 3x -7\\AC = x + 3x + 9 -7\\AC = 4x + 9 - 7\\AC = 4x+2$

We group x and 3x and add those together. Then we subtract 7 from 9.

With this equation, we can put in x = 8:

$AC = 4x +2\\AC = 4*8 + 2$

Since 4 * 8 = 32:

$AC = 4 * 8 + 2\\AC = 32 + 2\\AC = 34$

Finally, we have found both x and AC.

Answer

x = 8

AC = 34