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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