Simultaneous Equations

Today we had a look as solving simultaneous equations. These are equations that have 2 variables (letters, i.e. x, y, a, b, etc). Previously we have looked at solving standard equations that only have 1 variable, such as 2x + 6 = 10. We will be using these same techniques, but solving equations with 2 variables. If we are solving for 2 variables, we do need 2 equations to start with, so all of our problems with these will give us 2 equations.

The first thing we want to do when we are faced with simultaneous equations is to rearrange one of our equations so that it is "x = something" or "y = something". We do know how to do this from previous work, however I will run through a few examples.
Once we have done this, we can substitute this value for x or y into our other equation and solve this. This should give us a value of one of our variables. We can then use this value to find the other variable.

I know this all sounds complicated, however after a few examples I hope it should make more sense.

Example: Solve the following simultaneous equations

2y + x = 8   -> We will call this equation [1]
1 + y = 2x   -> We will call this equation [2]

First, we will rearrange equation [2] so we have "y = something". We will do this by getting rid of the 1 on the left hand side of this equation. This then gives us:

1 + y - 1 = 2x - 1
y = 2x - 1     -> We will now call this equation [3]

Now, we know that y = 2x - 1, so we can now take this knowledge and help us solve equation [1]. We then substitute (2x - 1) wherever there is a "y" in equation [1]. This is because equation [3] has told us that y = 2x - 1
We then get:

2(2x - 1) + x = 8

Now, expanding the brackets we get:

4x - 2 + x = 8

We can then collect like terms, and get:

5x - 2 = 8

Now solving this, we get:

5x - 2 + 2 = 8 + 2 
5x = 10
5x/5 = 10/5
x = 2

We now know that x = 2, so we have solved half of these equations. We can then substitute this value of x = 2 into equation [3], and solve this to get a value for y.
We the get:

y = 2(2) - 1
y = 4 - 1
y = 3

So now we know a value for y as well. So we have now successfully solved our equation. We have found that:
x = 2
y = 3


If you want further example of this, the following websites are good:

http://www.khanacademy.org/math/algebra-basics/core-algebra-systems/core-algebra-systems-tutorial/v/solving-linear-systems-by-substitution -> Watch the video, but only up to the 4min mark.

http://www.purplemath.com/modules/systlin4.htm -> This is handy, but some of the language they use and some of the examples are a little comple, but it still has some good information.


I have uploaded a new worksheet to the "Worksheets" tab. You should be able to work through the number style questions (i.e. question 1). Do not worry too much about the worded questions. I will go through those on Wednesday.

As always, if you have any questions feel free to comment on here, or email or chat to me in person.