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www.le.ac.uk
Equation of a Line
Department of MathematicsUniversity of Leicester
Contents
Using the Equation
Introduction
Forming the Equation
Parallel and Perpendicular Lines
Introduction
We are used to the equation for a straight line.
We’ll look at where this equation comes form, how to form it and what we can do with it once we’ve got it.
We are also going to see alternative ways of writing the same equation.
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Forming the Equation
Suppose we have:
How do we get the y-value from the x-value?
It’s a straight line, so the ratio between x and y values is the same for all values.
Eg. For this line, , or .
Ratio = = gradient, or m. We get .
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
m
1 2 3-1-2-3
1
2
-1
See what happens if you vary the value of m.
(Clear)
Forming the Equation
What if the line doesn’t go through the origin?
If the line starts at :
Then it’s ,
but with c added on to all the y-values.
So .Next
c
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
c
2 4 6-2-4-6
2
4
-2
See what happens if you vary the value of c.
(Clear)
But if we’re not starting at anymore, we can’t just find by finding the ratio of the values.
So instead, we use the ratio of the difference in value:
Forming the Equation
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Forming the Equation
If the line slopes down instead of up, the change in x will be negative:
So the gradient will be negative.
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
xy
2 4 6-2-4-6
2
4
-2
Try plotting an equation
(Clear)
What is the equation of this line:
Forming the Equation
1
2
1 2 3-1
-2
-1-2
3
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Forming the Equation
What is the equation of this line:
1
2
1 2 3
3
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Question: What is the equation of the line joining and ?
Answer: Start with ,
Then we know , so ,
so , or
Forming the Equation:Alternative ways of writing it
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
You could rewrite with everything on the same side of the equation:
And then if we have fractions, multiply through by the denominators to get
Eg.
Forming the Equation:Alternative ways of writing it
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Which of the following lines joins and ?
Forming the Equation:Alternative ways of writing it
All of them
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
If two lines are not parallel then they will have exactly one point of intersection.
You can find this by letting the 2 lines have the same - and -values, so they become 2 simultaneous equations…
eg.
The point of intersection is (-3,-3).
Using the equation: Intersection
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Where do and intersect?
Using the equation: Intersection
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
, Check answer Clear answers
Show answer and method
You can find the distance between two points using Pythagoras’s theorem:eg. Find the distance between (1,-2) and (3,1).
We get:
Then distance = hypotenuse= .
Using the equation: Distance
Next
(1,-2)
(3,1)
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
What is the distance between the points (3,6) and (-4,10)?
Using the equation: Distance
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Check answer
To find the midpoint of two points, you just find the average of the -coordinate and the average of the -coordinate.
So the midpoint of and is:
Using the equation: Midpoint
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
xy
What is the midpoint of (5,7) and (-3,11)?
Using the equation: Midpoint
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
, Check answer
Parallel lines will have the same gradient, because for the same change in , they both have the same change in .
So .
Parallel and Perpendicular Lines
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
For perpendicular lines, , and .
So .
Parallel and Perpendicular Lines
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
What is the gradient of the line parallel to ?
Parallel and Perpendicular Lines
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
Check answer
Which of the following lines is perpendicular to
?
Parallel and Perpendicular Lines
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines
A straight line has the equation:• , or • m = gradient, c = y-intercept
We can find the Intersection Point of 2 lines, and also the Distance or Midpoint between 2 points.
For parallel lines, .For perpendicular lines, .
Conclusion
Next
Using the Equation
Introduction Forming the Equation
Parallel and Perpendicular Lines