Mathematics - Geometry help solve geometry homework

 

I’m currently studying basic algebra and trying to get a clearer understanding of  and how it works in different situations. To practice, I’ve created a set of questions below. I’d like a detailed, step-by-step explanation for each part so I can learn how to solve similar problems on my own.

Part A

Given two points:
A(3, −2) and B(7, 4)

• Find the slope of the line passing through these points. 
• Clearly explain the slope formula and show each step of the calculation. 
• Describe what the slope means in this situation (for example: whether it is positive or negative, how steep the line is, and what direction it moves). 

Part B

Now suppose point B changes position to B′(7, −2).

• Calculate the new slope of the line connecting A(3, −2) and B′(7, −2). 
• Compare this slope to the one found in Part A. 
• Identify the type of line formed (horizontal, vertical, or diagonal) and explain your reasoning. 

Part C

Next, consider switching the y-values of the original points:

C(3, 4) and D(7, −2)

• Find the slope between these two new points. 
• Compare this slope to the original slope from Part A in terms of value and direction. 
• Finally, explain what happens if both points have the same x-value (for example: C(5, 4) and D(5, −2)). What kind of line would this produce, and why? 

I’m hoping to better understand how slope behaves under different conditions and how to recognize patterns when working with linear equations.