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Let  O be the origin. Points  P and  Q lie in the first quadrant. The slope of line segment OP is  1 and the slope of line segment  is  3. If  then compute the slope of line segment 

Note: The point  (x,y) lies in the first quadrant if both  and  are positive.

 Apr 7, 2024
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Understanding the Slopes:
 

Slope of line segment OP = 1: This means for every 1 unit you move up on the y-axis, you also move 1 unit to the right on the x-axis.

 

Since O is the origin (0,0), point P will be at (x, x) for some positive value x.

 

Slope of line segment PQ = 3: This means for every 3 units you move up on the y-axis, you move 1 unit to the right on the x-axis.

 

Finding Coordinates of P:

 

As mentioned earlier, P lies at (x, x).

 

Connecting P and Q:

 

We know the slope of PQ is 3. We need to find the y-coordinate of point Q relative to P (i.e., how much we move up from P). Let the y-coordinate of Q be y.

 

Relating Coordinates using Slope:

 

The difference in y-coordinates (y - x) is 3 times the difference in x-coordinates (which is just x in this case).

 

Setting up the Equation:

 

y - x = 3x

 

Combining Like Terms:

 

y = 4x

 

Finding the Slope of PQ:

 

Slope of PQ = (Change in y) / (Change in x) = (y_Q - y_P) / (x_Q - x_P)

 

Since both P and Q lie on the positive x-axis (first quadrant), x_Q and x_P are both positive values. Therefore, we can divide both sides of the equation y = 4x by x (assuming x is not zero) to get:

 

y/x = 4

 

This gives us the slope of line segment PQ, which is 4.

 Apr 9, 2024

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