Thursday, 1 December 2011

C1-Algebra and Functions - Solving Simultaneous Equations

We now,move on to Solving Simultaneous Equations. We will usually encounter questions where 1 equation is a quadratic and the other being a linear. From Gcse, you should know how to solve simultaneous equations by either Substituition or Elimination. Let's Recap :

1) y = 4x + 2
2) 3x + 6y =39

Let's Solve this firstly by Substituition:
* In General, we rearrange one of the equations to make x or y the subject, we then substitute this expression in the 2nd equation, to find one of the variables.
*Simply then plug into any equation, to find the value of the other variable.

We see y is already the subject of the first equation, let's substitute this equation as y in the 2nd equation so :
3x + 6(4x+2) = 39
Now expand and simplify:
3x + 24x + 12 = 39
27x =27
x= 1
Now Substitute this value of x=1, into any equation to find the value for y:
y=4(1) + 2

Solutions {x=1,y=6}

By Elimination
*In General, We usually eliminate one of the variables x or y, we do this by making the coefficient of the x's or y's same and the subtracting or adding them to make it 0.
*Now we only have one variable, so we solve for this variable. And substitute the value in one of the equations to get the value for the other variable.

1) y = 4x + 2
2) 3x + 6y =39

We can use either make the coefficient of x or y the same, we'll stick for y, it's easier in this case. Multiply the first equation by 6 :
6y = 24x + 12

Now rearrange the second equation to make 6y the subject :
6y = 39 - 3x

Subtract these two new equations :
       6y = 12 + 24x

   -   6y = 39 - 3x 
 which gives you :
0 = -27 + 27x
rearrange to solve for x:
27= 27x
x= 27/27 = 1

Now substitute this value in any of our two original equations for y :
y= 4(1)+2

Solutions = {1,6}

That shows how to solve two linear equations / simultaneous equations. To Solve a Simultaneous equation system with a quadratic, simply use Substituition :

1) Rearrange the the linear equation to make x or y the subject.
2) Substitute this expression for x/y, in the quadratic equation. Solve for this variable
3)Substitute the value you get, into either of the two original equations {linear or quadratic} to find the other variable's value.

I don't want to make this post too long, so i will add an example in my set of image notes.

Notes for Just "Simultaneous Equations"
Here is an example with Simultaneous with a quadratic involved:
Simultaneous With a Quadratic Equation

*Note - The example I have used kind of uses everything we learnt from the previous topics - Using how to solve a quadratic to manipulating surds. The answers are left in "Exact Values", i.e. not in decimals. Be sure that edexcel can test you in anyway way.