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.

Answer key here

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