To figure out the slope of a line when you have two points, there's a simple formula you can use. The two points will look like this: $(x_1, y_1)$ and $(x_2, y_2)$.

The formula for finding the slope ($m$) is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Breaking Down the Formula:

1. What the Terms Mean:

• $y_2$ and $y_1$: These are the $y$-values of the two points. The difference $(y_2 - y_1)$ shows how much the line goes up or down when moving from the first point to the second.
• $x_2$ and $x_1$: These are the $x$-values. The difference $(x_2 - x_1)$ tells us how much we move left or right between the two points.

2. The Slope Idea: The slope $m$ is often called "rise over run." Here, "rise" is the change in height (the $y$-values), and "run" is the change in distance (the $x$-values).

• A positive slope means the line goes up as you move from left to right.
• A negative slope means it goes down.

Example 1:

Let’s look at two points: $(1, 2)$ and $(4, 3)$.

To find the slope, follow these steps:

1. Identify the points:

   • Point 1: $(x_1, y_1) = (1, 2)$
   • Point 2: $(x_2, y_2) = (4, 3)$

2. Plug the values into the formula:

   $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{4 - 1}$$

3. Do the math:

   $$m = \frac{1}{3}$$

So, the slope of the line that connects the points $(1, 2)$ and $(4, 3)$ is $\frac{1}{3}$.

Example 2:

Now, let’s try with the points $(2, 4)$ and $(5, 8)$.

1. Identify the points:

   • Point 1: $(x_1, y_1) = (2, 4)$
   • Point 2: $(x_2, y_2) = (5, 8)$

2. Substitute into the formula:

   $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 4}{5 - 2}$$

3. Calculate:

   $$m = \frac{4}{3}$$

So, the slope for these points is $\frac{4}{3}$, which shows it has a steeper incline.

Important Points to Remember:

• If $x_2 - x_1 = 0$, you can't find the slope because you’re dividing by zero. This happens when both points have the same $x$-value, creating a vertical line.
• Always keep the order of the points the same. If you subtract $y_1$ first, do the same with $x_1$.

Learning how to understand and calculate slope is important in algebra. It helps you see how lines behave and their direction!