This checkpoint has 6 parts, each with their own video:

• Part 1 - The Basics
• Part 2 - Expansion, Factorisation and touching upon Surds
• Part 3 - Linear Equations
• Part 4 - Quadratic Equations
• Part 5 - Rearranging Equations
• Part 6 - Functions

Part 1 - The Basics

Algebra generally occurs in the form of:

• words
• or already expressed calculations.

In algebra the letters that you see are just representatives of an unknown number. They're called variables and you can also substitute in numbers there.

Algebra's kind of like a recipe, it shows you the steps but sometimes not the ingredients.

There are ways that algebra is expressed in its most simplest form (for ease). This is called simplification.

Here are some general rules:

• Addition – same variable, just add together. E.g. 2x + 5x + 2, x^2 + 4x + x
• Subtraction – same variable, just subtract. E.g. 5x – 2x + 1
• Multiplication – combine together. E.g 5 x x x 6x x 2y
• Division – see video.2x / 4x, 2x/5x. 2x/5y

You can even substitute in numbers (ingredients) to see if the formula/expression works (recipe).

• g. 2x + 5x + 2, x^2 + 4x + x

Working with equations

The key to solving equations is isolate out the variable. It pretty much involves using the opposite sign and moving things to the other side. Let me demonstrate with the following examples.

• 2o + 4 = 20
• o + 20 = 50
• o – 20 = 5
• 4x/10 = 4/5

Working with inequalities

Similar process as working with equations however:

When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality must change.
The solution of an inequality will be a range of values for the variable, rather than just one value.

• 2o + 4 < 20
• o + 20 > 50
• o – 20 < 5
• 4x/10 > 4/5

Now, let's go through some example questions:

Which inequation shows the following statement? x is greater than 7 but less than or equal to 10

A 10 ≥ x > 7 B 10 < x ≤ 7 C 10 ≤ x > 7 D 10 ≤ x ≥ 7 E 10 < x ≥ 7

If V= [(x-y)/4] then x equals

A Y + 4V B Y - V C Y/4 + V D Y + V/4 E 4(Y + V)

Part 2 - Expansion, Factorisation and touching upon Surds

Expansion

Expanding brackets involves removing the brackets from an expression by multiplying out the brackets. This is achieved by multiplying every term inside the bracket by the term outside the bracket.

Let's expand:

• 3(x+6)
• 6(4a−10).

Factorisation

Factorisation is the opposite process of expanding brackets. For example, expanding brackets would require 2(x+1) to be written as 2x+2. Factorisation would be to start with 2x+2 and to end up with 2(x+1).

Let's factorise:

• 2x+2
• 2x² + x - 3

To factorise:

• Take out common factors.
• Put the equation back together.

Surds

When we can't simplify a number to remove a square root (or cube root etc) then it is a surd!

Example: √2(square root of 2) can't be simplified further so it is a surd

Example: √4(square root of 4) can be simplified to 2, so it is not a surd!

Have a look at some more examples of simplifying surds:

• √2 x √4
• √100 x √4
• √2 + √4

Surds have a decimal which go on forever without repeating - they are actually Irrational Numbers.

Now, let's do some questions.

Q23 - Simplify 5√124 completely

A 10√31 B 7√31 C 20√31 D 10√62 E None of these

Q20 - Expand and simplify

7(x+2) + 5 =

A 7x + 19 B 7x - 19 C 7x - 9 D -7x + 19 E None of these

Q50 - Factorize and simplify a² – 6a + 9

A (a – 3)(a – 3) B (a + 3)(a – 3) C (a + 3)(a + 3) D (a – 3)(a + 3) E None of these

Part 3 - Linear Equations

The first part in understanding linear equations is knowing what a cartesian square is.

A cartesian square has a Y axis and X axis. It has spaces where coordinates can be placed. Below is a Cartesian square.

What are coordinates?

Coordinates are points on the Cartesian square which come in this form (x, y).

In the diagram above, you can see that in the top left hand quadrangle, that the coordinates are negative, positive. This means that any point there will always be negative and positive e.g. (-2, 5). This point is plotted as Point A as an example.

What’s it used for and why do you need to know it?

First, you need to know what it is because its concepts appeared many exams. Its underlying principles are used in a lot of things – for example, GPS systems, Google Maps and vector illustrations using in computer graphics. It is a big part of mathematics and is often referenced in other parts of mathematics, e.g. the unit circle in trigonometry.

What’s a linear equation?
It’s an equation, that is, something which equals something else which if plotted with y and x axes will make a straight line.

What’s its structure?
It follows this form:

2.3. Reconstructing a Linear Equation

The exam is likely to test your knowledge of coordinates and what the possible equation should be. This will draw from your knowledge of what the elements are that make up a linear equation.

They may give you a graph with two coordinate sets where you might have to reconstruct the linear equation. This is how you can do it quickly.

So, what do you do?

Assess the what you are given. You have a Cartesian Square with who lines. Each line, if they aren’t the same, would imply that there are two separate linear equations. We are also given the coordinates for one of the bold solid line.

Substitution

This is a strategy you can use to check answers and, in some cases, find answers giving the exam is a multiple choice one. This strategy isn’t just limited to linear equations, you can use it a lot, including using it for algebra and also quadratic equations.

Substitution is basically putting a number in for x to find out what the y number is.

Watch the next video (quadratic equations) as that will also cover questions in linear equations as the two are covered together.

Part 4 - Quadratic Equations

Now it's time to go through some questions:

Q 32 - Which equation could only be the equation of the graph?

A y = x + 2 B y = x - 2 C y = -x + 2 D y = -x - 2 E y = 2x - 2

Q 53 - The turning point of the graph could only be:

A (5, 6) B (-5, -6) C (-5, 6) D (5, -6) E (0, 5)

Q 55 - Find the point of intersection of the graphs of y = 4x + 5 and y = 3x - 9

A (14, 33) B (-14,-51) C (14, -51) D (-14, 33) E None of these

Part 5 - Rearranging Equations

Let’s say you get a question like this in the exam:

Make a the subject of b(a - ab) + 4(ab + a) = 2b

a) a = b(1 - b) ÷ 2b + 4
b) a = 2b ÷ (b(1 - b) + 4(b + 1))
c) a = (1 - b)(b + 1)
d) a = 2b(1 - b)(b + 4)

To answer this question, we need to get a on its own on one side of the equation.

To begin with, we can take a common factor of a out of the term b(a - ab) to get ab(1 - b).

We do this by dividing both terms by a (a ÷ a = 1, ab ÷ a = b), meaning that (a - ab) = a(1 - b).

Similarly, we can take a common factor of a out of the term 4(ab + a) to get 4a(b + 1). The left hand side of the equation now becomes ab(1 - b) + 4a(b + 1).

Next, we can take a common factor of a out of the entire left hand side of the equation. Both terms are multiplied by a, so if we divide out the a we get a(b(1 - b) + 4(b + 1)).

Our equation now reads a(b(1 - b)+ 4(b + 1)) = 2b.

Finally, since the left hand side of the equation is a term (b(1 - b) + 4(b + 1)) multiplied by a, we can divide both sides of the equation by this term to get a on its own. This gives us a result of a = 2b ÷ (b(1 - b) + 4(b + 1)), which is option (b). To recap:

1. Get a on its own to define the relationship between a and b
2. Simplify the expression as much as possible

Part 6 - Functions

Let’s say you get a question like this in the exam:

The function f(x)=x³+6x²+13x+2. What is f(2)?

a) 42
b) 60
c) 12
d) 83

This question requires us to understand and apply function notation. A function f(x) represents a relationship between x and y, and an expression like f(2) means f(x) for the value of x=2.

To find f(2), we need to substitute x=2 into the expression x³+6x²+13x+2. This gives us (2)³+6(2)²+13(2)+2, which we then need to simplify to find the value of f(2).

If we begin with the first term (2³), we know that this is equivalent to 2 x 2 x 2 = 23. We can begin with 2 x 2 = 4, and then multiply 4 by 2 one more time to get a result of 8. The next term is 6(2)², so we need to start by squaring 2 to get 4, and then multiply that by 6 to get 24.

After that, the next term is 13(2), which multiply to 26. Finally, we add 2.

Therefore, the equation now reads f(2) = 8 + 24 + 26 + 2.

Our answer is f(2) = 60, which is option (b). To recap:

1. Substitute x = 2 into the equation f(x)
2. Simplify to find the result of f(2)

Now it's time to do your assignment.

1. Download the assignment questions here.
2. Print it out or if you want to do it electronically, save it.
3. Complete the questions to it.
4. Then check the solutions on the video below. The answer key is also on the final page of your downloaded assignment questions.

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