The Point-Slope Form Equation Of A Line

Hi. The topic here is on the point slope form equation of a line. If the slope of a line is known and a point on a that line is known then this point slope form equation of the line can be used to help us obtain the y = mx + b form of the line.


Table Of Contents

The Point-Slope Form Of The Equation Of A Line

Examples

Practice Problems

Answers


The Point-Slope Form Of The Equation Of A Line

The point-slope form of the equation of a line with a point (x_{o}, y_{o}) is of the form:

    \[(y - y_{o}) = m(x - x_{o})\]

One may want to consider another version of the above equation. This is the version I was shown and taught back in my high school mathematics course a while back. The equation is:

    \[y = m(x - p) + q\]

where (p, q) is a point on the line given a slope m. This version replaces (x_{o}, y_{o}) with (p, q) and the q is added on both sides.

To obtain the y = mx + b form, algebra and the distributive law would be applied to either (y - y_{o}) = m(x - x_{o}) or y = m(x - p) + q.

Here is a visual on how the point and line interact.

Source: http://www.mathsisfun.com/data/images/graph-point-slope-a.gif


Examples

1) Given a point (-1, -5) on the line AB with a slope of 3. The equation of the AB line in the y = mx + b form is as follows:

Here we have m as 3 with y_{o} = -5 and x_{o} = -1.

Source: http://quicklatex.com/cache3/0f/ql_dd890d520cafcc7d25b3589a89a3230f_l3.png

 


2) Suppose that there is a point C (0, 2) and D (6, -4). What is the y = mx +b equation of the CD line?

First, we determine the slope of the CD line.

Source: http://quicklatex.com/cache3/be/ql_a287d9a7b922cad39fbe6da9e4d32abe_l3.png

 

Now, the point-slope form of the equation of the CD line can be used. This would help in obtaining the y = mx +b form. You can use either C (0, 2) or D (6, -4) as both points are on the CD line. I will be using point C (0, 2).

 

Source: http://quicklatex.com/cache3/14/ql_e0b28be2a7736425fedb44b58d583e14_l3.png

The equation of the line which passes through points C (0, 2) and D (6, -4) is y = -x + 2.


Practice Problems

1) Given a slope of -\dfrac{1}{2} and a point J (-2, 10), what is the equation of the line (y = mx + b form) which passes through point J?

2) Given a slope of 3 and a point K (7, -1), what is the equation of the line (y = mx + b form) which passes through point K?

3) What is the equation of a line in the y = mx + b form which passes through L (-2, 1) and M (5, 5)?


Answers

1) y = -\dfrac{1}{2}x + 9

2) y = 3x - 22

3) The slope is m_{LM} = \dfrac{4}{7}. The equation of the line is y = \dfrac{4}{7}x + \dfrac{15}{7}.

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