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Find the equation (given 1 point and slope)

Introduction

Every point in the coordinate plane can be part of many different lines.  Consider the point (2, 2) for example.  This point is part of one horizontal line, one vertical line, and an infinite number of diagonal lines. 

Find_eqn_given_1_pt__slope_vis_1

 

The diagram to the right shows several lines that go through the point (2, 2).  Each of these lines has a different slope.  Although the lines all intersect at the same point, each one has a different equation. 

Take a closer look at the red line.  It has a slope of 2.  In this lesson, you will learn how to find the equation of a line when given one point and a slope.


Lesson

 

Every line that can be drawn on the coordinate plane can be represented by an equation. 

When you know the slope of a line and the coordinates of a single point on the line, you can find the line’s equation.

 

 

The point-slope equation of a line is:   

Find_eqn_given_1_pt__slope_vis_10
 

 

The introduction to this lesson showed a number of lines that all went through the point (2, 2).  Example 1 demonstrates how to find the equation of the line that has a slope of 2 and goes though this point.

  

Example 1:  Find the equation of the line that goes through the point (2, 2) and has a slope of 2.

Find_eqn_given_1_pt__slope_vis_2Solution:

The equation is as follows:

Find_eqn_given_1_pt__slope_vis_3 

 


You can see from the graph and the slope-intercept equation that this line has a slope of 2 and a y-intercept of -2.  Most problems of this type can be done similarly.  Example 2 has a fractional slope, but the procedure is the same.

 

Example 2:  Find the equation of the line that goes through the point (6, 10) and has a slope ofFind_eqn_given_1_pt__slope_vis_4.

Solution:Find_eqn_given_1_pt__slope_vis_6a

 Find_eqn_given_1_pt__slope_vis_5
 

 One interesting aspect of the point-slope equation is that you are subtracting the values x1 and y1.  When the coordinate has two positive values, you actually subtract them when you put them into the equation.  Whenever you have a negative value for the coordinate (x1, y1), you will be subtracting a negative, which translates into adding the values in the equation.  This is shown in example 3.

 

 

 

Example 3:  Find the equation of the line that goes through the point (-4, -8) and has a slope ofFind_eqn_given_1_pt__slope_vis_7.

Solution:Find_eqn_given_1_pt__slope_vis_9a

  

Find_eqn_given_1_pt__slope_vis_8

 

 

Review the Concept

The process of finding the equation when given a point and a slope is pretty straightforward.  Simply use the equation for point-slope form:

 

Find_eqn_given_1_pt__slope_vis_10 

 

Once the slope and the coordinate have been put into the point-slope equation, solve the equation for y to change it into slope-intercept form.


Try It

 

Find slope-intercept equation of a line containing: 


1)  Point (2, 4) and slope -1 

2)  Point (-4, 2) and slopeFind_eqn_given_1_pt__slope_vis_11

3)  Point (0, 8) and slope 5

4)  Point (4, -9) and slope -3 

5)  Point (0, 0) and slopeFind_eqn_given_1_pt__slope_vis_12 

 

 

Scroll down for answers…

 

 

 

 

 

 

 

 

 

 

Answers:

1)  y = -x + 6

2)  y = -½x

3)  y = 5x + 8

4)  y = -3x + 3

5) y =Find_eqn_given_1_pt__slope_vis_12x


 

 

 

Related Links:

For more information on graphing lines and related topics, try one of the links below.

Resource Pages

 

Related Lessons

 

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