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Mathematical Formulae

Rearranging Equations I (Simple Equations)

Introduction

Before we can rearrange an equation we need to be clear about two things.  The first is the order in which we evaluate an expression and the second is to know which pairs of operations do and undo something, that is an operation and its inverse operation.

Evaluating Algebraic Expressions

First check to see if you have mastered order. Given an algebraic expression that involves a variable, say x, you must be able to evaluate the expression for a particular value of x. If you get this question right and feel confident, carry on with this page, otherwise you need to go to Order of operations first, master that then come back to this page.

Given the expression    y = 3x2 - 5, what is the order in which you would find y if you are given a value for x?

Multiply the value by 3, square the answer then subtract 5
Square the value then subtract 5 and finally multiply by 3
Multiply the value by 3, subtract 5 from the answer then square
Square the value, multiply the answer by 3, then subtract 5
Subtract 5 from the value, square the answer then multiply by 3

Inverse Operations

If you get this question right and feel confident, carry on with this page, otherwise you need to go to Inverse Operations first, master that then come back to this page.

For each operation choose the operation that will undo the operation on the left:

Add and
Divide and
Reciprocal and
Square root and

Rearranging Simple Equations

So now we have the skills needed to master rearranging equations.  The lone variable, usually on the left side of the equation, is called the subject of the formula.  In rearranging an equation, we are trying to make one of the other variables in the formula the new subject of the formula.

The rearranging process is much like dressing and undressing feet. 

Starting with bare feet, we first put on stockings, then put on shoes.

Take off undoes put on, but the order is important too.  Taking stockings off before shoes would be difficult to say the least.

Starting with shoes and stockings on, we first take off the shoes, then take off the stockings to end up with bare feet.

Mathematically, the process involves undoing all the operations to make say p from q in reverse order.to make q from p.

There are several ways of doing this. We will start with a pictorial way which can help you understand what is happening but is tedious to draw, then use some of the shorter ways which should make sense after that.

Let's go back to the expression    y = 3x2 - 5   At the moment y is the subject of the formula.  We are going to make x the new subject.  So we start with x and perform the operations in order to make y.  Then we start with y and undo each operation in reverse order to get x.

Start with x, square it, multiply by 3, then subtract 5 to get y. Reversing the process, starting with y we add 5, divide by 3, take the square root and this is x.

So we get the new expression x=square root of (y+5/3) .

Let's just check it works. If we take the original expression    y = 3x2 - 5 and set x = 6 then
y = 3(6)2 - 5 = 3(36) - 5 = 108 - 5 = 103. 

When we set y = 103 and use the new expression:
x=sqrt((y+5)/3) = sqrt((103+5)/3)=sqrt(108/3)=sqrt(36)=6

Great, we got the expected value for x!

Often we use a much abbreviated form of this to rearrange an equation.

With practice you will not need to add the simplify steps and may even be able to rearrange several steps in your head.  The most important thing is to remember is reverse order of inverse operations.

You may remember the equation for converting a Fahrenheit temperature to a Celcius temperature.

C=5/9*(F-32)

Let's rearrange this to make F° the subject of the formula.  So we start with F°. We can see from the brackets that the first operation to do is subtract 32, then multiply by 5 and finally divide by 9 to get C°.  So we need to start with C°, multiply by 9, divide by 5 then add 32 to get F°.

F=9*C/5 + 32

What is the Fahrenheit temperature when the temperature is -10°C?     

-10°C = °F   

More Exercises

 

<< Inverse Operations and Functions | Mathematical Formulae Index | Rearranging Equations II (Quadratic Equations)>>

 

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