Finding the Value of x: A Midpoint and Linear Equations Problem

In this problem, we are given the following information:

• x is the midpoint of the line segment yz
• The equation xy = 6x + 4 is provided
• The equation yz = 40 is provided

Our goal is to find the value of x.

Step 1: Understand the Concept of Midpoint

The midpoint of a line segment is the point that is equidistant from the two endpoints of the segment. In other words, if we divide the line segment into two equal parts, the midpoint would be the point that separates these two parts.

Step 2: Utilize the Midpoint Formula

Since x is the midpoint of yz, we can use the midpoint formula to express the relationship between x, y, and z. The midpoint formula is:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

In our case, the midpoint is x, and the endpoints are y and z. Therefore, we can write:

x = ((y + z)/2, (y + z)/2)

This means that the x-coordinate and the y-coordinate of the midpoint x are both equal to (y + z)/2.

Step 3: Solve the System of Equations

Now, we have two equations and two unknowns (y and z):

xy = 6x + 4 yz = 40

We can substitute the midpoint formula into the first equation to get:

((y + z)/2)y = 6x + 4

Simplifying this equation, we get:

(y^2 + yz)/2 = 6x + 4

We can then use the second equation, yz = 40, to solve for y in terms of z:

y = 40/z

Substituting this into the first equation, we get:

(40^2 / z^2 + 40)/2 = 6x + 4

Solving this equation for z, we get:

z = 20

Now, we can substitute z = 20 into the equation yz = 40 to find y:

y = 40/20 = 2

Finally, we can use the midpoint formula to find the value of x:

x = ((y + z)/2, (y + z)/2) x = ((2 + 20)/2, (2 + 20)/2) x = 11

Therefore, the value of x is 11.