Lines

Lines, parallel, perpendicular, slope intercept, point-slope.

Introduction

As was mentioned in the last section, lines are one of the geometric object that requires the least amount of information to distinguish them. The only information we need is two points. From this information we can find the slope of the line. From here we can determine if lines are two lines are parallel or perpendicular. Then there are two common ways to write the equation of a line. These are called the slope intercept form and the point slope form.

Slope, Parallel, Perpendicular

As was mentioned before the first thing we will discuss is the slope of a line. Given two points, $(x_{1},y_{1}){\text{ and }}(x_{2},y_{2})$ the formula for the slope is ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$. From a geometric standpoint the slope tells you how quickly the line increases in y-value per each unit change in x-value.

Example: Find the slope of the line containing (5, 3) and (6, 9).

The slope, denoted m, is ${\frac {9-3}{6-5}}=6={\frac {3-9}{5-6}}$.

Two lines $y=m_{1}x+b_{1}{\text{ and }}y=m_{2}x+b_{2}$ are parallel if $m_{1}=m_{2}$

For example, y = 5x + 3 is parallel to y = 5x - 4, but is not parallel to y = -3x + 8.

Two lines, $y=m_{1}x+b_{1}{\text{ and }}y=m_{2}x+b_{2}$ are perpedicular if $m_{1}={\frac {-1}{m_{2}}}$.

For example, $y=3x-2$ is perpendicular to $y={\frac {-1}{3}}x+5$, but is not parallel to $y=3x+5$

Slope intercept Form

The slope intercept form for a line provides us with both the slope, and y intercept without requiring any work. It also allows us to write the equation of a line with only two pieces of information, the slope and y-intercept. Given a slope m, and y-intercept (0, b) the slope intercept form for the line is y = mx + b.

For example, Given a line with slope 5 and y-intercept (0, -2) the equation of the line, in slope intercept form, is y = 5x - 2.

Point Slope Form

The point slope form is a more widely applicable way to write the equation of a line. Just like the slope intercept form we only require knowledge of the slope, but we only need a point that is on the line, even if it is not the y-intercept. Given a point $(x_{1},y_{1})$ and a slope m, the point slope form for a line is $y-y_{1}=m(x-x_{1})$.

Example: Find the equation of a line with slope -2 going through the point (3, 7).

Solution: The equation for the line is $y-7=(-2)(x-3)$

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