Lines in the Plane and Slope

Introduction

The simplest mathematical model for relating two variables is the linear equation $y=mx+b$ (Slope-intercept form). This equation is called Linear because its graph is a line. $m$ is the slope and $(0,b)$ is the y-intercept.

Finding the slope $m$

For instance, suppose you want to find the slope of the line passing through the distinct points $(x_{1},x_{2})$ and $(y_{1},y_{2})$.

 $Slope={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {y_{1}-y_{2}}{x_{1}-x_{2}}}$

Exercises Find the slope of the line passing through the distinct points below

1) $(-6,2)$ and $(3,20)$

Solution:
$2$

2)$(3,-7)$ and $(-3,-7)$

Solution:
$0$

3)$(3,-2)$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-3,1)}

Solution:
${\frac {-1}{2}}$

Writing the linear equation given a slope and a point on the line

 Point-Slope Form of the Equation of a Line
 The equation of the line with slope  passing through the point $(x_{1},y_{1})$ is 
 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y-y_{1}=m(x-x_{1})}

Notice: In order to write this equation, we need a point and a slope given

Exercises Find the equation of the line line given the information below

1) slope Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m=3} and goes through $(1,2)$

Solution:
Apply the formula with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m=3} , $x_{1}=1$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{1}=2} to get the result
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y-2=3(x-1)}

Writing the linear equation given two points on the line

Given two point $(x_{1},y_{1})$ and $(x_{2},y_{2})$ are on the line. To find the equation of this line:

First, use the formula to find the slope

Then, apply the point-slope formula with the slope we just found and one of the given points.

Exercises Find the equation of the line passing through the distinct points below

1) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (4,3)} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,-5)}

Solution:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m=slope={\frac {3-(-5)}{4-0}}={\frac {8}{4}}=2}
Apply the point-slope formula with slope Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m=2} and the given point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (4,3)} ( I choose $(4,3)$ in this case, but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,-5)} will give the same result) to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y-3=2(x-4)}

Notes:

A vertical line goes through has equation of the form $x=a$ where $a$ is any constant.

This page were made by Tri Phan