1. Department of Mathematics University of Leicester
Equation of a Line
Department of Mathematics
University of Leicester

2. Parallel and Perpendicular Lines
Contents
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines

3. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using 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.
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4. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using 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
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5. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
See what happens if you vary the value of m. 1 2 3 -1 -2 -3
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6. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
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 . c
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7. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
See what happens if you vary the value of c. 2 4 6 -2 -4 -6
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8. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Forming the Equation
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:
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9. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using 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.
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10. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Try plotting an equation 2 4 6 -2 -4 -6
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11. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Forming the Equation
What is the equation of this line: 1 2 3 -1 -2

12. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Forming the Equation
What is the equation of this line: 1 2 3

13. Forming the Equation: Alternative ways of writing it
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Forming the Equation: Alternative ways of writing it
Question: What is the equation of the line joining and ?
Answer: Start with ,
Then we know , so ,
so , or
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14. Forming the Equation: Alternative ways of writing it
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Forming the Equation: Alternative ways of writing it
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.
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15. Forming the Equation: Alternative ways of writing it
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Forming the Equation: Alternative ways of writing it
Which of the following lines joins and ?
All of them

16. Using the equation: Intersection
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Using the equation: Intersection
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).
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17. Using the equation: Intersection
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Using the equation: Intersection
Where do and intersect?
Solve the simultaneous equations:

18. Using the equation: Distance
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Using the equation: Distance
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=
(1,-2)
(3,1)
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19. Using the equation: Distance
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Using the equation: Distance
What is the distance between the points (3,6) and (-4,10)?

20. Using the equation: Midpoint
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Using the equation: Midpoint
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:
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21. Using the equation: Midpoint
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Using the equation: Midpoint
What is the midpoint of (5,7) and (-3,11)?

22. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
Parallel lines will have the same gradient, because for the same change in , they both have the same change in . So
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23. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
For perpendicular lines, , and So
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24. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
What is the gradient of the line parallel to ?
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25. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
Which of the following lines is perpendicular to ?

26. Parallel and Perpendicular Lines
Introduction
Forming the Equation
Using the Equation
Parallel and Perpendicular Lines
Conclusion
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,
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