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.
[tex]AC = AB + BC[/tex]
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.
[tex]AB = BC[/tex]
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...
[tex]AB = x+9\\BC = 3x-7[/tex]
...we can put these values into our 2nd equation and solve for x:
[tex]AB = BC\\x + 9 = 3x -7[/tex]
Add 7 to both sides:
[tex]x + 16 = 3x[/tex]
Subtract x from both sides:
[tex]16 = 2x[/tex]
Divide both sides by 2:
[tex]8 = x\\x = 8[/tex]
Knowing x, we can find the distance of AC using our first equation.
[tex]AC = AB + BC[/tex]
Let's put in the values of AB and BC:
[tex]AC = (x+9) + (3x-7)[/tex]
Before we put in x = 8, we can simplify this:
[tex]AC = (x+9) + (3x-7)\\AC = x + 9 + 3x -7\\AC = x + 3x + 9 -7\\AC = 4x + 9 - 7\\AC = 4x+2[/tex]
We group x and 3x and add those together. Then we subtract 7 from 9.
With this equation, we can put in x = 8:
[tex]AC = 4x +2\\AC = 4*8 + 2[/tex]
Since 4 * 8 = 32:
[tex]AC = 4 * 8 + 2\\AC = 32 + 2\\AC = 34[/tex]
Finally, we have found both x and AC.
Answer
x = 8
AC = 34
