How can I find the equation of a line using two given points?

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To find the equation of a line given two points, let’s denote the points as (x1, y1) and (x2, y2). The steps to derive the line’s equation—commonly expressed in slope-intercept form (y = mx + b)—involve calculating the slope and then using one of the points to find the y-intercept (b). Here’s how you can do it:

  1. Calculate the slope (m): The slope of the line is determined by the formula:


    m = (y2 - y1) / (x2 - x1)


    Substitute the coordinates of your points into this formula. The slope represents how steep the line is.
  2. Use the slope-intercept form: The slope-intercept form of a line is written as:

    y = mx + b


    Plug the slope (m) you just calculated into this formula. Now, you need to find the y-intercept (b).
  3. Find the y-intercept (b): To do this, you can use one of the two points. Let’s use (x1, y1);

    Substitute its coordinates into the equation:


    4. y1 = mx1 + b


    Rearranging this gives:


    b = y1 - mx1

  4. Write the equation of the line: Now that you have both the slope (m) and the y-intercept (b), substitute them back into the slope-intercept form:


    y = mx + b


    This is the equation of the line that passes through your two given points.

Example: Say the points you have are (2, 3) and (4, 7). First, calculate the slope:

  • m = (7 - 3) / (4 - 2) = 4 / 2 = 2

Now, use the slope with one of the points (we’ll use (2, 3)) to find b:

  • 3 = 2(2) + b ➔ 3 = 4 + b ➔ b = 3 - 4 = -1

So the equation of the line is:

  • y = 2x - 1

This process lets you smoothly derive an equation from any two points—just remember to follow each step logically!

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