# Generating Equation of the Line

Line is one of the three undefined objects in plane geometry. This time, we will tackle line in coordinate plane. I will provide sample problems and solution to understand deeply how to generate equation of a line out of the given conditions.

Forms of equation:

Standard form:  $\displaystyle{Ax+By+C=0}$

Slope intercept form: $\displaystyle y=mx+b$ where $\displaystyle{m}$ is the slope of the line and $\displaystyle{b}$ is the y-intercept

Sample Problem 1: Given two points

Find the equation of the line through (-1,3) and (4,-2).

Solution:

To solve this type of problem we need to look for the slope of the line first

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$ $\displaystyle m=\frac{-2-3}{4+1}$ $\displaystyle m=\frac{-5}{5}$ $\displaystyle {m=-1}$

And we select one of the of the given points let’s say $\displaystyle (-1,3)$

Using the Point-Slope formula,

$\displaystyle y-y_1=m(x-x_1)$ $\displaystyle y-3=-1(x+1)$

$\displaystyle x+y-2=0$ is the desired equation of the line in standard form.

Sample Problem 2: Given Slope and y- intercept

Find the equation of the line with slope 2 and y-intercept of 2.

Solution:

Using the equation $\displaystyle{y=mx+b}$ we can easily identify the equation of the line

By substitution:

$\displaystyle y=2x+2$

$\displaystyle 2x-y-2=0$ is the required equation in standard form

Sample Problem 3: Given x and y intercepts

Find the equation of the line with -2 and 3 as x and y intercepts respectively.

Solution:

Using the formula,

$\displaystyle\frac{x}{a}+\frac{y}{b}=1$ where a & b are x and y intercept respectively.

By substitution,

$\displaystyle\frac{x}{-2}+\frac{y}{3}=1$ $\displaystyle 3x-2y=-6$ $\displaystyle 3x-2x+6=0$

Practice Problems:

1. Find the equation of the line through (2,1) and whose slope is twice the slope of the line 3x-2y-4=0.

2. Find the equation of the line through (-1,2)  with 4 as its x-intercept.

3. Find the equation of the line through (2,3) and (-3,1).

4. Find the equation of the line with 3 and -4 as its x and y intercepts respectively.

5. Find the equation of the line with slope of $\frac{1}{2}$ and x-intercept of 3.