The point-slope form of a linear equation is an important way to understand how two things relate to each other in algebra. It helps us describe a straight line using its slope and a specific point on that line.

The basic formula looks like this:

$$y - y_1 = m(x - x_1)$$

In this formula:

• $(x_1, y_1)$ is a point on the line.
• $m$ is the slope of the line.

The slope ($m$) shows us how steep the line is. We can think of it as how much the $y$ value goes up or down (the "rise") for every change in the $x$ value (the "run").

This formula is really helpful because if you know the slope and a point on the line, you can quickly draw the line.

Example

Let’s say you know a line goes through the point (2, 3) and that the slope is 4.

If we plug these values into the formula, we get:

$$y - 3 = 4(x - 2)$$

This form can also help us find other ways to write the linear equation, like the slope-intercept form and the standard form.

Changing Point-Slope Form to Slope-Intercept Form

The slope-intercept form looks like this:

$$y = mx + b$$

Here, $b$ tells us where the line crosses the $y$-axis (the $y$-intercept).

Using our earlier example, we can change the equation from point-slope form to slope-intercept form. Starting with:

$$y - 3 = 4(x - 2)$$

First, we need to simplify it. We do this by distributing the 4:

$$y - 3 = 4x - 8$$

Next, we add 3 to both sides to solve for $y$:

$$y = 4x - 5$$

Now we see that the slope $m$ is 4, and the $y$-intercept $b$ is -5. So, we have successfully rewritten our equation in slope-intercept form!

Changing to Standard Form

The standard form of a linear equation looks like this:

$$Ax + By = C$$

In this case, $A$, $B$, and $C$ need to be whole numbers, and $A$ should be positive. To convert from slope-intercept to standard form, we start with:

$$y = 4x - 5$$

1. First, move $4x$ to the left side:

   $$-4x + y = -5$$

2. Next, multiply everything by -1 to keep the numbers positive:

   $$4x - y = 5$$

Now our equation $4x - y = 5$ is in standard form.

How to Use Point-Slope Form

The point-slope form is very useful. It’s great when you have a point and want to quickly find the equation of the line.

It’s also helpful for graphing. You can start with the known point and use the slope to find more points on the line.

For example, if your slope is 2 and you start at the point (1, 1), you could go up 2 units (the rise) and then over 1 unit (the run), reaching the point (2, 3). By doing this a few times, you can plot more points and draw the line.

Quick Summary of Conversions

Here’s a quick way to remember how to change between these forms:

• From Point-Slope to Slope-Intercept: Distribute, isolate $y$, and find $b$.
• From Slope-Intercept to Standard Form: Rearrange to the form $Ax + By = C$ and make sure the numbers are whole.

It’s important for students to practice moving between these forms. Understanding and using the point-slope form lays the groundwork for dealing with different types of linear equations.

As students become more skilled in these conversions, it will help them with more challenging topics later, like systems of equations and functions. It also shows how useful math is in real life when we want to model relationships. Mastering the point-slope form makes learning algebra more manageable and relevant!