Slope and Equation of a line

Slope and Equation of a line

In business and in many technical reports, the graph is best picture that tells the reliable information. Basically, when the slope of the graph tilts upward, it talks positive, and negative otherwise. Slope is very important in the field of dynamic mathematics, statistics, business, economics, and a lot that involves data application. In the field of mathematics especially calculus, the slope formula is basically the need. To find the slope of a line, any two points should have a specific location in its graph, else setting y and x to zero might help.

Illustration:

The line cannot be told if it is not in the "Cartesian"plane. Cartesian plane looks like in the figure i below.

figure i

This is the empty one. Lines and its slope will be drawn here. Note that Cartesian plane varies in looks and colors sometimes but in concrete positioning, that is, x and y axes cannot be interchanged and their corresponding graduation as well.

Points of a line should be found in;

,  ordered pair,

and the general equation of a line passing through a point is;




Finding a slope is simply having;


, m is a symbol we use as the slope of a line. Also, the rise/run in the graph.The value of such is the slope of a line after putting values to x's and y's. So far, all authors uses it.


The point-slope formula

The  figure below shows if the slope is negative or positive.

Illustrative Examples:

1. Find the slope of a line given the ordered pairs (0 , 3), (-2 , 2) and (6 , 6), (-1, 5/2).

So using any pair can reveal the slope of this line. Let'rs use (0 , 3), (-2 , 2) .

Given:

                 

             

Solution:

 

- this means we have 1 as rise and 2 as run. Since the rise is 1 and the run is 2, added these to the point (0,3), it gives us (0+2,3+1) or (2,4), This predicts the possible "other" point location of the line. See figure ii below (colored in blue).

figure ii:

We highlighted to focus the locations of ordered pairs from the figure ii above.

2. Find the equation of a line with the following information;

,

The general equation of a line is  . so

So the line based from the given is found to be;

Now applying the point-slope formula;  

 , this is now the equation of the line.

3.) Find the equation of a line with slope 3 @ point (1, 4 ).

Given;

, 

                 

Solution:

 , this is now the equation of the line.

4.) Find the equation of a line given point (2,-3) and (6,1).

Getting the slope first;

Using first point  (2,-3) to find "b" in the general equation of a line.

So, substituting the value of 'b' in the general equation of a line, we found that;

 , or simply 

 , this is now the equation of the line.

Let's try using point (6,1) to find the equation of the line. Let's see if they have the same with point (2,-3). Let's have the same process.

Getting the slope first;

Using first point  (1,6) to find "b" in the general equation of a line.

substituting the value of 'b' in the general equation of a line,

 , or simply

 , this is now the equation of the line. They arrived at the same result!

Example Problems:

1.) Find the equation of a line at points (1/2 , 3) and ( 5, 3), and  show its graph.

Getting the slope first;

Using point (1/2 , 3) to find 'b'.

Now, substituting the value of 'b' in the general equation of a line,

 ,  Answer, this is the exact location of 'y' in the graph.

Let's try using second point (5, 3) to check if they have the same result.

@ m = 0, then finding 'b' is;

Now, substituting the value of 'b' in the general equation of a line,

 , or simply

  , this is the exact location of 'y' in the graph. Note that they arrived at the same result! Answer

...and  show its graph

Now, the graph of the line @ (1/2 , 3) and ( 5, 3) as our Answer

2.) Given the graph below, find the equation of the line.

Here, we can easily identify the "b", it crosses to y-axis @ 2, so our b=2. If we check the dotted lines, it tells the location of our slope m, the rise (numerator) 4 and the run (denominator) is 3. So our slope m is found to be 4/3. Thus the equation of out line is simply substituting these newly-found values.  

So, 

 , this is the equation of the line presented by the graph above!

3.)From the given equations below,  determine if the given lines are parallel or perpendicular nor neither. 

*Parallel lines have the same slope and never intersects.

*Perpendicular lines are lines with negative reciprocals of each other, these lines intersects with 90 degrees.

Given equations :

Solving each line;

line 1:

  ,   

line 2:

  ,  

Notice that they have the same slope m, which is 4/3.thus they are parallel.

4.)  Determine if the given lines are parallel or perpendicular nor neither. See equations below.

*Parallel lines have the same slope and never intersects.

*Perpendicular lines are lines with negative reciprocals of each other, these lines intersects with 90 degrees.

Given equations:

Solving each line;

line 1: 

 , m = 3

line 2: 

 , m = -1/3

Here, we have two different slopes, 3 and -1/3, they are negative reciprocals. Thus the lines are perpendicular.

5.) Solve and show the graph of the line given its equation below,

Solution:

Set x and y to zero to find the values of each intercepts.

@ x = 0                         @ y =0

             

                           

The graph:

Practice Test: (with partial solution)

Find the equation of the lines given their following details, or provide what is asked;

1.) points (2,3), and (3, 2)

2.) m = -2/3 , @ x = 4

3.) graph where the points are @ (-2, 3), and m = -3

4.) what is the slope of a line y = 3x- 1?

5.) show the graph of the slope 2/3 @ point (3, 4) 

Solution #1:

click for answer

Solution #2:

click  for answer

Solution #3:

the graph

      

solution #4:

click  for answer


solution #5: 

the graph

Simple Quiz:

Find the equation of the lines given their following details, or provide what is asked;

1.) points (2/3,5), and (-1, -4) , click 

2.) m = 1/2 , @ x = 3 , click 

3.) graph where the points are @ (-2/3, 3/5), and m = -3/4 , click 

4.) what is the slope of a line y = 2/5x- 3/7? , click 

5.) show the graph of the slope -1 @ point (-4, 5) , click 

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