You can put this solution on YOUR website! OP is the entire line and the point M lies somewhere along the line between the endpoint O and the endpoint P.
.
The sum of the lengths OM and MP equals the entire line OP. You are told that the length OM is:
.
.
and the length MP is:
.
.
Therefore, the some of these two line segments is the sum of the right sides of these two equations. In other words, the equation form of the sum of these two segments is:
.
.
On the right side of this equation, the parentheses can be removed without changing the terms they contain because both sets of parentheses are preceded by a plus sign. This makes the equation become:
.
.
Combine the two constants on the right side and the two terms containing x as follows:
.
.
.
.
You are also told that the length of the line OP (as represented by OM + OP) is 44. From that you can write that the right side of this equation equals 44 as follows:
.
.
Get rid of the +2 on the left side by subtracting 2 from both sides:
.
.
and this simplifies to:
.
.
Solve for x by dividing both sides by 3 to get:
.
.
.
Now that we know the value of x is 14, you can find the length of OM (given by x+8) and the length of MP (given by 2x-6). Substitute 14 for x in both these and you get:
. which simplifies to
.
and
. which first simplifies to and then to
.
Note that the two line segments OM and MP have lengths of 22. Since they are equal in length, by the definition of midpoint as the point that divides a line into two equal segments, you can say that the point M is the midpoint of the line OP.
.
Hope this helps you to understand the problem and the way that you can solve it.
(