Solving different kinds of equations

General Information
“Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns." Is the only bullet point under Equations and Expressions, Number and Algebra of Level 6 Mathematics and Statistics. It is arguably the main focus of the Algebra exams, mainly assessed directly prior to the 2016 exam, but following the dramatic change in style of the 2016 exam, it is now assessed mostly in the form of worded questions, the difficulty being not only in solving the equations, but also in coming up with them. Note that use of an equation in any question is required for any mark higher than Achieved in that question.

Linear Equations/Inequations:
Linear equations are equations in which there is only a single unknown, to the power of at most one. The main skills involved in solving linear equations/inequations are simplifying by combining like terms and by altering both sides of the equation.

Your first step is to make any calaulations necessary in order to put everything containing your unknown (x,y,n...) on one side of the equation, and the rest on the other side. Keep in mind that all alterations made to one side of the equation must also be made to the other side. For example: in the equation x+5=34, we subtract 5 from the left side to isolate x, and get x+5-5. We then do this to the other side, and get x+5-5=34-5. x=29 in this case.

Then, multiply/divide both sides by a certain number so that your unknown is one times itself. You now have the unknown's value. 5x=30  5x/5=30/5    x=6

Inequations work in the same way, except with <, >, ≤ or ≥ in place of the =. However, when dividing or multiplying by a negative number, always reverse the sign, so that and ≤ becomes ≥.

Exponential equations:
An exponential relationship occurs when the y values change by a constant multiplier. Each y value is multiplied by the same number.

The graph shows that y values are doubled. They are multiplied by 2 each time, this makes it an exponential relationship.

As they are multiplied by 2 each time, we can predict that the next value is 16.

Quadratic Equations
Quadratic equations are equations where the highest power of x (the unknown number) is x². The way to solve quadratic equations is more complicate than solving linear equations. Quadratic equations very commonly have 2 different answers. (notable differences are equations where x2±2ax+a2=0)

The way to solve a quadratic equation is to move all the numbers to one side, which means the other side must be 0 (even if one side ends up negative!). On your first step, you have to make sure the equation has factorised out the multiples. In this equation: 3x²-12x+15=0, we can factorise 3 out of the equation and it will become 3(x²-4x-5)=0 (x2-4x-5=0/3, x2-4x-5=0). Now, we must solve x2-4x-5=0. We can factorise this and we will get (x+1)(x-5). The method is to make 5 into 5×1 first, then we can see the -5 on the left which means there are 1 negative and 1 positive. Let 5×1 become two separate number 1 and 5, we know one number is positive one number is negative and it has to end up -4. So we make 5 become -5 and 1 stays positive, when we add up these two numbers together it becomes -4, that means we are correct. Put the number in to brackets with x to form (x-1)(x-5). Because the other side of the equation is 0, one of the brackets must be 0, so we get x=1, x=5 as our two answers. In a real-life situation, select the appropriate answer (usually the positive).

Here is a guide for you for numbers and algebra:

https://numbers-and-algebra.wikia.com/wiki/Numbers_and_Algebra_Wiki?action=edit&section=2