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Complex linear equations
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Simultaneous linear equations
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What are simultaneous linear equations, and what are they for?

They are pairs (initially, eventually you can have lots and lots of them!) of equations which apply to the same data. This is the beginning of mathematics starting to be really useful in that real world that so many of our students like to ask us about.

In this introduction, I am going to talk about how they work, what they do for us and what they mean, which all contend for the things that confuse maths students: first prize.

Let's start with a simple puzzle. It is much simpler to make sense of things, if you have a sense of the need for them before we start.

In a cafe, 3 cups of tea and a bun cost £1.60 and 2 cups of tea and a bun costs £1.25. What is the cost of a cup of tea and what is the cost of a bun?

Let's begin by reading the problem and then we'll see how to translate the problem into mathematics.

1. Let $1$ cup of tea cost $t$ pence and
2. Let $1$ bun costs £$b$.
3. So 3 cups of tea and a bun will cost £$3t+b$.
4. And 2 cups of tea and a bun will cost £$2t+b$.
5. Now, because we know what these combinations cost, we can state our sentences in maths as follows: \begin{align} 3t+b&=1.6\\ 2t+b&=1.25 \end{align}
6. We can see our problem restated as a pair of simultaneous linear equations, which, once we learn to solve them, will enable us to find out the cost of the single cup of tea and also the cost of the bun.
7. The solution to this problem is that $t=0.35$ and $b=0.55$. If you substitute those values into the equations, you should see that they come out right.
8. $3(35)+55=1.6$
9. $2(35)+b=1.25$

Solving the equations is beyond the scope of this introduction, but this is what we will be looking at in this topic section.

In mathematics, in order to solve an equation with one unknown, we need only one equation. If, as in the example above we have $2$ variables, we need $2$ equations. If you have $3$ variables, you need $3$ equations and so on...

A useful aspect of working with two equations in 2 unknowns is that we can graph them. I have used the $x$ axis to show the cost of a cup of tea and the $y$ axis to show the cost of a bun. When we draw the lines on the graph, we can see that there is only one point on the graphs where both lines match up and that is the solution we are after. The point is marked on the graph and shows that $t=0.35$ and $b=0.55$.