Download Notes for 1A: Solving Equations In Your Head

Notes for 1A: Solving Equations In Your Head
What is a pronumeral?
A pronumeral is any letter or symbol (usually a letter) used to represent a number. So far you are
familiar with pronumerals used to represent unknown angles from the Geometry Unit. You are
also familiar from primary school with finding missing numbers represented by boxes. These
boxes are also pronumerals, except in high school we usually use letters to represent the missing
numbers instead.
If we are going to use pronumerals in mathematics, then we need to be careful that the letters and
symbols we use don't look like symbols we already use for numbers or operations or any other
mathematical symbol. The most obvious case is the letter x and the multiplication sign !.
To avoid confusion, mathematicians always draw the letter x with curly parts, and the
multiplication sign with straight parts. When handwriting, the letter 'z' can look like the number 2
so many mathematicians draw strike a line through the middle of the z, like we do with the
number 7 (to avoid getting confused with the number 1).
What is an equation?
An equation is a mathematical statement that tells us that two things are equal.
For example:
2 ! 4 = 11 – 3
Note that both the Left Hand Side and the Right Hand Side equal 8.
What is an algebraic equation?
An algebraic equation is an equation with one or more pronumerals in the equation.
For example:
2x + 1 = 7
We are used to using equations to write the answer to some expression. For example, when
evaluating 2 ! 3 + 1, we find the answer is 7 and write 2 ! 3 + 1 = 7. Algebriac equations
however, often tell us the answer, but have some missing number in the expression. In the example
above, the missing number was represented by the pronumeral x.
What does it mean to solve an algebraic equation?
Algebraic equations have some missing number in them. Solving an algebraic equation simply
means to find the value of this unknown number so that the equation will be true. To be true
means that the two sides are actually equal. 2 + 2 = 5 – 1 is true. But 2 + 2 = 1 is false.
Solving an equation requires us to work out the missing number and to write the value of it
underneath.
For example: x + 5 = 7
The answer is x = 2 since this is the only value for x that will make the equation true.
How do you solve equations?
There are several methods you can use to work out the mystery number. One is simply doing it in
your head by thinking about what number would make it work.
For example: x + 2 = 7. I just think about it like this. “What number plus 2 equals 7”. By
thinking backwards, it should be obvious the answer is 5.
The other ways we use to solve equations are (i) backtracking and (ii) doing the same to both
sides. This latter method (along with doing it in your head for easier examples) is the main
method used in secondary school.
What does 2x and 3x and 4x mean? There's a number and a letter next to each other!!!
If we want to multiply the number 2 with the pronumeral x, instead of writing 2 !x, we can just
write 2x. In mathematics, between any number and letter there is an invisible multiply sign.
So if you see 2x or 3x, it just means 2 ! x or 3 ! x etc. Or, in practical terms, it means 2 lots of x
or 3 lots of x etc.
Example:
4x means 4 ! x. (In practical terms, this means 4 lots of x)
Please note, you will not see anything like x2 or x4 anywhere in this algebra topic. That is
because this isn't allowed. If you want to simplify x ! 9, you must say 9x, not x9. The rule is THE
NUMBER GOES FIRST, THE PRONUMERALS come after.
Example:
Rewrite t ! 7 in a simpler way.
The rule is NUMBER FIRST, PRONUMERAL COMES AFTER.
So it's 7t.
What does
x
x
or
mean? A fraction with pronumreals in it???
2
3
Instead of writing x ÷ 2 or x ÷ 3, we often use fractions to represent the same thing. So x ÷ 2 is
x
x
represented as
. And x ÷ 3 is represented as
. Fractions mean “top divided by bottom”.
2
3
Example:
Represent y ÷ 5 as a fraction.
Answer:
y
5
The same also works when we want to divide by a pronumeral, but then the pronumeral goes on
the denominator.
Example:
Represent 10 ÷ p as a fraction
Answer:
10
p
Finally, you can combine both a multiplication and a division to form one fraction.
Example:
Represent m ! 7 ÷ 2
Answer:
7m
2
7
m . Pronumerals on the
2
numerator can either be put as part of the numerator or put to the side.
Both representations mean exactly the same thing!
Note: This could also be represented as
How do I solve things in my head?
If the equation is an addition of 2 things, we think backwards. We are actually using the same
process as we use with subtraction.
Example.
x + 5 = 9.
I think what plus 5 equals 9.
I do “5 less than 9” in my head.
The answer is x = 4.
If the equation is a subtraction problem, I again think backwards. So I am actually using the same
process as I use with addition.
Example:
x – 6 = 12.
I think what take 6 equals 12.
I do “6 more than 12” in my head.
The answer is x = 18.
Example:
10 – x = 7.
I think what do I take from 10 to get 7.
I do “from 7, how much more to 10” in my head.
The answer is x = 3.
If the equation is a multiplication problem, I think backwards. So I am actually using the same
process as I use with division.
Example:
3x = 12.
I think what multiplied by 3 equals 12.
I do “how many times does 3 go into 12” in my head
The answer is x = 4.
If the equation is a division problem, it may be represented as a fraction. To make the fraction
equal the number on the other side of the equation, the numerator must be that many times bigger
than the denominator. For example, if we want the fraction to equal 7, then we need the
numerator to be 7 times bigger than the denominator.
Example:
y
!7
4
I think y needs to be 7 times bigger than 4 to make the fraction equal 7
I do “what is 7 times 4” in my head
The answer is y = 28
Example:
18
!6
x
I think 18 needs to be 6 times bigger than x to make the fraction equal 6.
So I think x needs to be 6 times smaller than 18
I do “6 goes into 18, how many times” in my head
The answer is x =3.
If the equation has multiple parts, I might not work out what x is straight away but instead work
out what larger parts are.
Example:
2x + 1 = 5.
I see it as 2 things added together.
I work out what the '2x' part means by thinking what plus 1 equals 5
I realise the first part must be 4.
Now I know 2x = 4.
Now I think “what multiplied by 2 equals 4”.
The answer is x = 2.
Example:
" 3x # 2 $
!5
4
I see that the fraction needs to equal 5.
I know the numerator must be 5 times larger than 4.
So the bracket, (3x + 2) must equal 20.
I see 2 parts in the bracket.
I think what plus 2 equals 20.
So the first part, 3x, must equal 18.
I think 3 times what equals 18.
So x must equal 6. The answer is x = 6.