To solve equations using the balance method, you need to ìunwrapî the target variable
one step at a time using inverse operations.
And the ìbalanceî part is that whatever you do on one side of the equal sign, you
also have to do on the other.
You have to keep the equation in balance, by doing the same operation on both sides
each step.
The key to getting the balance method right is good setting out.
You must be very careful to set things out neatly and correctly, then you wonít go wrong,
and even if you do, itíll be very easy to find and correct your mistake.
* Always work down the page.
Work vertically, not horizontally.
* Always keep equal things underneath each other.
And * Always line up your equal signs directly
underneath each other, down the page.
Always work down, not across.
Letís have a go.
Solve x minus sixty-seven equals eleven.
Now, yes, I know this is a very simple equation, and you can probably just guess the answer
by inspection.
But I want to use it to show you how the setting out works, down the page, keeping equal things
underneath each other.
And remember to make sure your ìxî looks like a letter, not a times sign.
Start by writing the equation out, with plenty of space between the left hand side and the
equal sign.
Donít be afraid of putting in extra white space; it will help you to keep things lined
up and see clearly what youíre doing.
I recommend putting the equal sign about a third to a half of the way across the page,
at least to begin with, until you get used to knowing how much room to leave for different
kinds of equations.
Okay, letís solve this equation using the balance method.
We have to ìunwrapî the variable, x, by doing inverse operations.
Well, the only operation here is a subtraction: ìminus sixty-sevenî.
The inverse of that is to add sixty-seven.
So letís add sixty-seven to both sides.
Remember, you have to keep it balanced: if I add sixty-seven to the left hand side, to
cancel out the minus sixty-seven, then I also have to add it to the right hand side.
Eleven plus sixty-seven is seventy-eight, so x must equal seventy-eight.
Notice the way all the equal signs are lined up vertically, one under another, and the
way Iíve written things that are same as on the line above directly underneath.
Thatís important.
It helps you to keep things organised, and to find your mistakes if you make any.
Letís try the next one.
Solve three p minus six equals eighty-four.
First, write the equation out, with plenty of space around it.
Again, put the equal sign some distance over so you have room to write the steps in where
you need to.
Now, I have to ìunwrapî that variable, p.
If I was working forwards, substituting a value in for p, I would first be multiplying
by three, and then subtracting the six.
So to go backwards, to unwrap the p layer-by-layer, I have to unwrap the minus six first, before
unwrapping the times three.
So, first: undo the minus six by adding six to both sides.
(Keep it balanced by doing the same thing on both sides.)
That leaves me with three p on the left, and eighty-four plus six, which is ninety, on
the right.
Now, undo the three times p by doing the inverse operation: divide by three on both sides.
Take two lines to write your fractions neatly and vertically (or you can use a divide sign
instead, if you prefer).
p equals thirty.
I can check my answer by going back to the original equation and substituting in my answer
wherever I see a p.
Three times thirty is ninety, minus six is eighty-four.
Good.
When you get better at this, you may be tempted to skip a few steps.
You can do that, but realise it will make it harder to find your mistakes.
And it also means youíre not communicating as well exactly what youíre doing at each
step.
Please, donít just write the ìplus sixî and ìdivide by threeî on the same line.
Some people even write it really small, on the outside of both sides.
Both of these methods lead to other problems later on, when the equations youíre solving
are not this simple.
Always write out a new line for a new step, and always write proper mathematical notation
on every line.
Iíll show you what I tend to write, as an experienced mathematician.
You see Iím still writing the same number of steps, but Iím not writing as much as
the full working I had before.
But itís still clear at each step what Iím doing to work my way towards the solution:
add six, then divide by three.
This way of setting it out turns the ìbalancingî into almost more of a ìtransferî from one
side to the other.
You just have to remember to always do the inverse operation when you transfer it to
the other side.
I recommend you stick with the full setting out first, and only consider writing less
than this when you feel absolutely confident that you know what youíre doing.
Next one: five plus four y equals thirty-seven.
So, write it out with plenty of space around it.
We have to unwrap the variable, y, by doing inverse operations in the opposite order to
normal.
But donít be confused by the order in which itís written.
Normally Iíd be doing the times by four first, and then adding the five.
So Iím going to have to subtract five first, and only then divide by four.
Subtract five from both sides.
Four y equals thirty-two.
Note itís important to ìsubtract fiveî.
You canít just write ìfive minusî on the front.
You have to think through the operations, and the order theyíre being done; and you
have to do the same thing to both sides.
You want to subtract five because that will cancel out the five at the front: five minus
five is zero.
Now divide by four, to get y equals eight.
And check your answer by substituting it back into the original equation.
Five, plus four times eight, is five plus thirty-two, which is thirty seven.
Good.
What if the expression is a bit more complicated?
Hereís one with some brackets in it.
The process is exactly the same; just think carefully about what order to use to unwrap
the variable.
In this equation, normally I would do the brackets first, and then the times by six
out the front.
So to go backwards to get to the variable, I have to divide by six first, which gives
me a plus nine equals twelve, and then undo the plus nine from inside the brackets: a
equals three.
Again, check the answer by substituting it back into the original equation.
Careful, do the brackets first.
Three plus nine is twelve, and six times twelve is seventy-two.
Final example: r over six, minus five, equals three.
Well, normally Iíd do the divide by six first, then the minus five.
So letís unwrap it by adding five first: r over six equals eight.
Now times by six to cancel the divide: r must equal forty-eight.
Check the answer: forty-eight over six is eight, minus five is three.