1. Equations of Lines in Another Format
Slideshow 31, Mathematics
Mr Richard Sasaki, Room 307

2. Objectives
Understand how to draw a line to an equation in the form 𝑎𝑥+𝑏𝑦+𝑐=0 by making 𝑦 the subject.
Understand how to draw a line to an equation in the form 𝑎𝑥+𝑏𝑦+𝑐=0 by connecting two pairs of co-ordinates

3. Drawing Graphs
We now know how to draw a graph based on two simple pieces of information:
The line’s gradient
The y-intercept it passes through
This is great when the function is in the form 𝑦=𝑎𝑥+𝑏 but not when it’s in a different form!

4. Changing The Subject
Another common way to write down the name of a line is in the form 𝑎𝑥+𝑏𝑦+𝑐=0. But now it’s not so clear what the gradient or y-intercept is…so what should we do? That’s right, change the subject!

5. Changing The Subject
Let’s try changing 𝑎𝑥+𝑏𝑦+𝑐=0 (the general form) so that 𝑦 becomes the subject.
𝑎𝑥+𝑏𝑦+𝑐=0⟹
𝑎𝑥+𝑐=−𝑏𝑦
⟹𝑦=− 𝑎𝑥 𝑏 − 𝑐 𝑏
So in this form, the gradient is − 𝑎 𝑏 and the constant is − 𝑐 𝑏 . Sounds messy doesn’t it…
Anyway, let’s try changing the subject with numbers!

6. Example Draw the graph 3𝑥+2𝑦−4=0. Let’s change it so 𝑦 is the subject.
3𝑥+2𝑦−4=0⟹
3𝑥−4=−2𝑦
⟹𝑦=− 3𝑥
∴𝑦=− 3𝑥 2 +2
The gradient is − 3 2 and the y-intercept is 2. Let’s draw it!

7. Example 3𝑥+2𝑦−4=0 (𝑦=− 3𝑥 2 +2) So it passes through y=2…
And it’s gradient is − So we go right two and down three to meet our line again.
Something like this…

8. Answers
Shown are not ideal titles but the equation where y is the subject.
𝑦=𝑥+2
𝑦= 3 2 𝑥+3
𝑦=− 1 2 𝑥+ 1 2
𝑦=−6

9. Finding co-ordinates
If we are given a line 𝑎𝑥+𝑏𝑦+𝑐=0, can you see an easy pair of co-ordinates that we could find?
When written in this form, it’s easy to find the two points…
Because when 𝑦=0, we get 𝑎𝑥+𝑐=0…
And when 𝑥=0, we get 𝑏𝑦+ 𝑐=0. We can then easily find 𝑥 and 𝑦 at those points!

10. Example
Draw the line 16𝑥+4𝑦−8=0 by finding two suitable pairs of co-ordinates.
Let’s find co-ordinates 𝑥, 0 and (0, 𝑦) where 𝑥 and 𝑦 are numbers.
When 𝑥=0…
4𝑦−8=0
4𝑦=8
𝑦=2
So we get (0, 2).
When 𝑦=0…
16𝑥−8=0
16𝑥=8
𝑥= 1 2
So we get ( 1 2 , 0).

11. Example This time we are drawing a line! What do we do? 16𝑥+4𝑦−8=0
(Note: Last time when we had two pairs of co-ordinates we used 𝑎= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 because we were calculating the line.
This time we are drawing a line! What do we do?
16𝑥+4𝑦−8=0
Simply place our co- ordinates , 0 and (0, 2) onto the graph and join the dots!
Note: You must use a ruler!!

12. Answers
The square would have co-ordinates (5, 0), (0, 5), (-5, 0) and (0, -5) and would have 4 line segments 𝑦=𝑥+5, 𝑦=𝑥−5, 𝑦=−𝑥+5, 𝑦=−𝑥−5.
3, 0 𝑎𝑛𝑑 (0, 3)
0, 6 𝑎𝑛𝑑 (−2, 0)
−2, 0 𝑎𝑛𝑑 0, 1 2
0, 9 𝑎𝑛𝑑 (−4, 0)