When you study linear equations in Grade 12 Algebra I, you will often come across something called the point-slope form. Knowing what this form is and how it works can make writing and using linear equations much easier. Let’s explore the point-slope form together!

What is Point-Slope Form?

The point-slope form of a linear equation looks like this:

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

In this equation:

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

This form is super useful when you already know a point on the line and the slope. You can easily write an equation without needing to make any complicated changes.

Important Features

1. Simple to Use with Given Points: If someone gives you a point $(x_1, y_1)$ and the slope $m$, you can quickly plug them into the formula. For example, if the point is (2, 3) and the slope is 4, you can write the equation like this: $y - 3 = 4(x - 2)$

2. Flexible for Graphing: The point-slope form helps you write the equation in a way that makes it simple to graph the line. You're working directly with the slope and a point, which is helpful when making a graph.

3. Can Change to Other Forms: While point-slope form is easy to use, you can also change it to other forms like slope-intercept form ($y = mx + b$) or standard form ($Ax + By = C$). For the example above, if we expand the equation: $y - 3 = 4(x - 2)$ $y - 3 = 4x - 8$ $y = 4x - 5$ It becomes slope-intercept form!

4. Understanding the Line Quickly: Looking at the point-slope form lets you see the slope and a specific point right away. This makes it easier to understand how the line behaves. For instance, if the slope $m$ is positive, then the line goes up from left to right.

5. Vertical Lines: One thing to remember about point-slope form is that it doesn’t work for vertical lines. Vertical lines have an undefined slope, so they can't be written in this form. Instead, they are shown like this: $x = a$, where $a$ is the x-coordinate of any point on that line.

Example to Understand

Let’s say you have a line that goes through the point (1, 2) and has a slope of -3. You can write the point-slope equation like this:

$y - 2 = -3(x - 1)$

If you want to quickly graph this line, start at the point (1, 2). From there, you can find more points by going down 3 units and right 1 unit again and again.

Conclusion

The point-slope form of linear equations is a key tool in algebra. It makes writing equations from a point and a slope pretty simple. By understanding how this form works, students can effectively solve and graph linear relationships. Learning to use point-slope form can greatly improve your skills with linear equations!