1) AB + BC = AC. Plug in for each of these, you get:
x - 3 + 7x + 12 = 6x + 23
Combine like terms:
8x + 15 = 6x + 23
Subtract 6x from both sides and subtract 15 from both sides so you get variables on one side and constants on the other.
8x - 6x = 23 - 15
2x = 8
Divide both sides by 2 to isolate the variable.
x = 8/2 = 4
x = 4.
2) Same thing to start out. AB + BC = AC.
2y + 5 + 3y - 4 = 7y - 5
Combine like terms:
5y + 1 = 7y - 5
Subtract 5y from both sides and add 5 to both sides so that you get variables on one side and constants on the other.
1 + 5 = 7y - 5y
6 = 2y
Divide both sides by 2 to isolate the variable.
y = 6/2 = 3
y = 3.
Now we must plug in our y-value. If AB = BC, then point is the mid-point.
2y + 5 = 3y - 4
2(3) + 5 = 3(3) - 4
6 + 5 = 9 - 4
11 ≠ 5
Point B is not the midpoint.
3) AH + HS = AS
5b - 1 + 12b + 7 = 8b + 9
Combine like terms:
17b + 6 = 8b + 9
Subtract 8b from both sides and 6 from both sides to get variables on one side and constants on the other:
17b - 8b = 9 - 6
9b = 3
Divide by 9 on both sides to isolate the variable:
b = 3/9 = 1/3.
Now, we have to find AS. So plug in your b-value into the AS equation.
AS = 8b + 9
8(1/3) + 9
8/3 + 9 -> 8/3 + 27/3 = 35/3 = 11 and 2/3
AS = 11 and 2/3 units