Other Equations of Lines: Algebra Chapter 2 Section 5

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  This entry is focusing a little deeper into the concept of lines, especially the other equations of the line.

Suppose that we have a line with slope $m$ and it passes through a specific point $(x_{1}, y_{1})$.  For any other general point (x, y) on the line (where $x_{1} \neq x$ and $y_{1} \neq y$) on the line, the slope of the line would be $m=\frac{y-y_{1}}{x-x_{1}}$.  Multiplying both sides of the equation by the denominator of the right hand side gives us $m \times (x-x_{1})=(y-y_1)$.  This final form is called the point-slope form of the linear equation, since we use a point and the slope.

The two green lines are parallel.
If we look at two parallel lines, they do not cross (they have no solution) by the definition of parallel lines.  This can only be done if the change in the y variable (the rise) for every unit change in the x variable are exactly the same for both lines.  Notice that "the change in the y variable (the rise) for every unit change in the x variable" is the definition of the slope.  So what this means is that two lines are parallel if and only if the two slopes are the same.  And yes, that means that two undefined slopes (vertical lines) are parallel.

The two red lines are perpendicular.  As are the y-axis and the x-axis.
Looking at perpendicular lines, we see that the point where they meet form nothing but right angles (angles of 90 degrees).  This occurs when, for two lines with slopes $m_{1}$ and $m_{2}$, $m_{1} \times m_{2}=-1$, which means that the slope of one line is what is called "the negative inverse" of the other slope.  Since a vertical line has undefined slope, we do have to specify that a horizontal line and a vertical line are also perpendicular to one another, since they do form right angles.

That's the end of this section.  If you have any questions, please leave them is the comments.  Like and share this post if you found it helpful.  And until next time, stay curious.

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