How can I determine the slope and the y-intercept of a given line?

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To find the slope and the y-intercept of a line, you typically start with the equation of the line in the slope-intercept form, which is:

y = mx + b

In this equation:

  • m represents the slope of the line.
  • b is the y-intercept, the point where the line crosses the y-axis (where x is 0).

Here’s how to find both:

1. Identifying the Slope (m)

The slope is a measure of the steepness of the line. It can be calculated using the formula:

m = (y2 – y1) / (x2 – x1)

Here, (x1, y1) and (x2, y2) are any two points that lie on the line. Simply plug in the coordinates of these points to find the slope.

2. Finding the Y-Intercept (b)

The y-intercept is found directly from the equation of the line. If you are given a standard form equation like:

Ax + By + C = 0

You can convert it to slope-intercept form:

By = -Ax - C
y = (-A/B)x - (C/B)

From this, you can see b = -C/B. This value is the y-coordinate where the line crosses the y-axis.

Example

If you have a line defined by the points (2, 3) and (4, 7):

  • First, calculate the slope:
    • m = (7 - 3) / (4 - 2) = 4 / 2 = 2
  • Next, use one of the points to find the y-intercept. Plugging in (2, 3) into the equation:
    • 3 = 2(2) + b
3 = 4 + b
b = 3 - 4 = -1

So, the slope is 2, and the y-intercept is -1.

Now you can express the line’s equation as:

y = 2x – 1

In summary, the slope and y-intercept are crucial for understanding the behavior of a line on a graph. With practice, identifying these values becomes a straightforward process!

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