1. Linear and nonlinear relationships
Chapter 4
Linear and nonlinear relationships

2. In this case, (-2, -4) is the solution of these simultaneous equations
Previously…
We had been looking at single linear graphs:
Sketching a linear graph using various methods
Finding the equation of a linear graph
Finding the distance between two points
Finding the midpoint between two points
Now we are looking at what happens when there are two linear graphs – known as simultaneous equations because they occur together at the same time
What we are ultimately trying to find is the point at which these lines cross or intersect – this is known as the simultaneous solution
We can find this solution graphically (by sketching the two graphs and reading the point of intersection), or algebraically by using substitution.
Note: if the gradient of the two lines is the same (same ‘m’ value, or the number in front of the x in y=mx+c), then the lines are parallel and have no solution as they never intersect
In this case, (-2, -4) is the solution of these simultaneous equations

3. Ex 4A – graphical solutions of simultaneous equations
To solve simultaneous equations graphically, it means reading the point of intersection off the graph
For this, it is important that the graph is accurate (use graph paper and plot the points carefully) – a rough sketch won’t do
Sometimes the question will give you the graph and you just have to read the point of intersection. Other times you will need to draw the graph yourself to read the point of intersection.

4. Question 1(a)
Use the graph to write the point of intersection (i.e. the solution to the simultaneous equations)
The equations for these two lines are:
x + y = 3
x − y = 1
Looking at the graph, we can see the point of
intersection is here…
This coordinate is (2, 1), which is our answer!
You can CHECK this by substituting x=2 and y=1 into the equations given at the start
x + y = 3
2 + 1 = 3  this gives us the correct answer so we know it is right!
x – y = 1
2 – 1 = 1  this is also correct, so we can be confident we have read the graphs correctly

5. Question 3(a) For equation: 2x + y = 8… For equation: x + y = 5…
Solve each of the following simultaneous equations using a graphical method
This means you have to sketch the two graphs first (using whatever method you like from exercise 3A – I recommend the x and y intercept method)
You should use graph paper for this as your graphs need to be super accurate in order to read the point of intersection correctly
For equation: x + y = 5…
To find x-intercept, let y=0
x + 0 = 5
x = 5
Plot point (5, 0)
To find y-intercept, let x=0
0 + y = 5
y = 5
Plot point (0, 5) and draw the line connecting the points
For equation: 2x + y = 8…
x-int: let y=0
2x + 0 = 8
2x = 8
x = 4
Plot point (4, 0)
y-int: let x=0
2(0) + y = 8
y = 8
Plot point (0, 8) and draw the line connecting the points
Now that both lines are sketched, write the point at which they intersect…
Which is (3, 2)

6. Question 4
Question 4: Solve each of the following pairs of simultaneous equations
This question uses the same steps as question 3, it’s just that the equations are a bit more complicated
Use whatever method you like to sketch the two graphs, then find the point of intersection
Use previous question as a guide
However, there is another way…

7. Alternative method for question 4
Then we can use algebra to find what x is…
8 − x = x + 2
(we need the x’s to be on the same side of the = sign)
8 = x + x + 2
(now simplify)
8 = 2x + 2
(now we want the 2 to go to the other side)
8 – 2 = 2x
(simplify)
6 = 2x
(now get x on its own)
6 ÷ 2 = x
3 = x
Then, put x = 3 back into one of the equations from the start to get the y value
y = x + 2
y = 3 + 2
y = 5
Therefore our solution is (3, 5)
You can use substitution rather than drawing the graphs for question 4
Let’s look at 4(a)
y = 8 − x y = x + 2
Both of these equations are equal to y, therefore we can say that:
y = y
8 − x = x + 2
You cannot use this method if the graphs don’t equal the same thing

8. Question 7 (reasoning) Jet ski question
Here are some hints to help you:
When it says ‘write the rules relating to the cost and length of rental’, it means you need to write two equations in the form y = mx + c
Is the ‘y’ the time, or the cost? (what value goes up the y-axis?)
Is the ‘x’ the time, or the cost? (what value goes along the x-axis?)
What is the ‘c’ part of the equation based on the information given? (what is the initial price you pay even if you haven’t had any time on the jet ski yet?)
Sketch the two graphs and use the evidence from these graphs to answer questions c and d

9. Ex 4E – linear inequations
See for video tutorial and worked examples from this exercise
An ‘inequation’ just means a normal equation (e.g. 2x + 3 = 10), but instead of an equals sign (=), it uses an inequality sign (either <, >, ≤, or ≥)
< is ‘less than’
> is ‘greater than’
≤ is ‘less than or equal to’
≥ is ‘greater than or equal to’
For the most part, you can treat these inequality signs just like a normal equals sign when you are working out what x is by undoing the things that have been done to it
The only exception is if you are multiplying or dividing by a negative number – in this case, you flip the direction of the inequality sign (< becomes >, and ≤ becomes ≥ and vice versa)
Refer to your notes from exercise 2D (solving linear equations) for more examples

10. Ex 4F – sketching linear inequations
Linear inequations replace the equals sign in a typical equation (y=mx+c) with an inequality sign (>, ≥, < and ≤)
The graph of linear inequations is a half plane (meaning it shades in half of the Cartesian plane)
The line that divides the plane is the line of the inequation
Consider the linear inequation y ≥ x + 2. There are many points (x, y) that satisfy this inequation – i.e., anything in the shaded area
What if it was y < x + 2? The solutions would be everything below the line (as it uses the ‘less than’ sign ‘<’)
Use a solid line if it is ≥ or ≤ to show that the numbers on the line are included in the solutions
Use a dotted line if it is < or > to show that the numbers on the line aren’t included in the solutions

12. Steps
y ≥ x + 2
y = x + 2
Sketch this linear graph using the x and y intercept method
To find the x-intercept: let y=0
0 = x + 2
x = 0 – 2
x = -2
x –intercept is (-2, 0)
To find the y-intercept: let x=0 or read it off the equation (y = mx + c where c is the y- intercept)
y-intercept is (0, 2)
As it uses a ‘greater than or equal to’ sign, connect the points with a solid line.
Shade the required region – in this case, what is above the line
Write the question
Write the linear equation form (i.e. swap the inequality sign for an = sign)
Sketch this linear graph using the x and y intercept method
To find the x-intercept: let y=0
To find the y-intercept: let x=0 OR read it off the equation (y = mx + c where c is the y-intercept)
Look at the original question – what sign does it use? If it is < or >, connect these points using a dotted line. If it is ≥ or ≤ connect these points using a solid line.
Shade the required region – either above the line (if > or ≥) or below the line (if < or ≤)
Done!

13. Let’s do Ex 4F question 5
Happy Yaps Dog Kennels charges $35 per day for large dogs (dogs over 20 kg) and $20 per day for small dogs (less than 20 kg). On any day, Happy Yaps Kennels can only accommodate a maximum of 30 dogs.
If l represents the number of large dogs and s represents the number of small dogs. Write down an inequation, in terms of l and s, that represents the total number of dogs at Happy Yaps.
Note: don’t be fooled! This question is just looking at the number of dogs, nothing about the cost!
Total number of dogs (regardless of whether they are small or large) must be less than or equal to 30 at any time (so we know which sign to use)
Answer: l + s ≤ 30
Another inequation can be written as s ≥ 12. In the context of this problem, write down what this inequation represents.
What was s?
Small dogs
So this inequation means that the number of small dogs must be 12 or more (or, at least 12)

14. Think back: what do we know?
The inequation l ≤ 15 represents the number of large dogs that Happy Yaps can accommodate on any day. This inequation is shown as a bold line on the graph, clearly shade in the area that is not within the region for this inequation.
Explore the maximum number of small and large dogs Happy Yaps Kennels can accommodate to receive the maximum amount in fees.
Think back: what do we know?
The fee is $35 a day for large dogs
The fee is $20 per day for small dogs
On any day, they cannot have more than 30 dogs total
They cannot accept more than 15 large dogs
They must have at least 12 small dogs
So, to get the highest fees, they want to maximise the number of large dogs as they cost more. So, 15 large dogs. That leaves room for 15 small dogs to be at capacity and to get the largest income.