Algebra

Introduction to Algebra – Basic Definitions

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What is an Equation?

An equation says that two things are equal.

It will have an equals sign “=” like this: x+ 2 = 6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)

Parts of an Equation

People can talk about equations, there are names for different parts (better than saying “that thingy there”!)

Here we have an equation

A Variable is a symbol for a number we don’t know yet. It is usually a letter like x or y. A number on its own is called a Constant.A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient) An Operator is a symbol (such as +, ×, etc) that represents an operation (ie you want to do something with the values).
A Term is either a single number or a variable, or numbers and variables multiplied together. An Expression is a group of terms (the terms are separated by + or – signs)

Polynomial

Example of a Polynomial: 3x2 + x – 2

A polynomial can have constants, variables and the exponents 0,1,2,3,…

And they can be combined using addition, subtraction and multiplication, … but not division!

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x2) are the same.

In other words, terms that are “like” each other. (Note: the coefficients can be different)

Example: -2xy, 6xy2 , (1/3)xy2    

Are all like terms because the variables are all xy2

 

What is a Formula?

A formula is a special type of equation that shows the relationship between different variables.

(A variable is a symbol like X or V that stands in for a number we don’t know yet).

Example: The formula for finding the volume of a box is: V = wdh

V stands for volume, w for width, d for depth and h for height.

  If w=5, d=10 and h=4, then V = 5×10×4 = 200

A formula will have more than one variable. The following are all equations, but only some are formulas:

x = 2y – 7 Formula (relating x and y)
a2 + b2 = c2 Formula (relating a, b and c)
x/2 + 7 = 0 Not a Formula (just an equation)

 

Subject of a Formula

The “subject” of a formula is the single variable that everything else is equal to.

Example: in the formula: s = ut + ½ at2

s” is the subject of the formula

Changing the Subject

One of the very powerful things that Algebra can do is to “rearrange” a formula so that another variable is the subject.

Rearrange the volume of a box formula (V = wdh) so that the width is the subject:

  • Start with:
V = wdh
  • divide both sides by d:
V / d = wh
  • divide both sides by h:
V / dh = w
  • swap sides:
w = V / dh

So now if you have a box with a depth of 2m, a height of 2m and a volume of 12m3, you can calculate its width:

w = V / dh

w = 12m3 / (2m×2m) = 12/4 = 3m

Substitution

In Algebra “Substitution” means putting numbers where the letters are:

Example 1: If x=5 then what is 10/x + 4 ?

Put “5” where “x” is:

10/5 + 4 = 2 + 4 = 6

Example 2: If x=3 and y=4, then what is x2 + xy ?

Put “3” where “x” is, and “4” where “y” is:

32 + 3×4 = 9 + 12 = 21

Example 3: If x=3 (but you don’t know “y”), then what is x2 + xy ?

Put “3” where “x” is:

32 + 3y = 9 + 3y

(that is as far as you can get)

As that last example showed, you may not always get a number for an answer, sometimes just a simpler formula.

Negative Numbers

When substituting negative numbers, put () around them so you get the calculations right.

Example 4: If x = -2, then what is 1-x+x2 ?

Put “(-2)” where “x” is:

1 – (-2) + (-2)2 = 1 + 2 + 4 = 7

 

Introduction to Algebra – Multiplication and Division

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Please read Introduction to Algebra – Addition and Subtraction first

A Puzzle

What is the missing number?

The answer is 2, right? Because 2 × 4 = 8.

Well, in Algebra we don’t use blank boxes, we use a letter. So we might write:

which can simply be written like this “Put the number next to the letter to mean multiply

You would say in English “four x equals eight”, meaning that 4 x’s make 8. And the answer would be written:

How to Solve

On the previous lesson we showed this neat step-by-step approach:

  • Work out what to remove to get “x = …”
  • Remove it by doing the opposite
  • Do that to both sides

It still works, but you have to know that dividing is the opposite of multiplying. Have a look at this example:

We want to remove the “4

To remove it, do the opposite, in this case divide both sides by 4:

Which is … Solved!

Just remember …

What you do to one side of the “=” you should also do to the other side!

Another Puzzle

Solve this one:  x / 3 = 5

  • Start with:
x / 3 = 5
  • What you are aiming for is an answer like “x = …”,
  • So, let us have a go at multiplying both sides by 3
x / 3 x 3= 5 x 3
  • A little arithmetic (3/3 = 1 and 5×3 = 15) becomes:
= 15
More Complicated Example

How would you solve this?  x / 3  + 2 = 5

It might look hard, but not if you solve it in stages.

First let us get rid of the “+2“:

  • Start with:
x/3 + 2 = 5
  • To remove the plus 2 use minus 2 (because 2-2=0)
x/3 + 2 -2 = 5 -2
  • A little arithmetic (2-2 = 0 and 5-2 = 3) becomes:
x/3 + 0 = 3
  • Which is just:
x/3 = 3

Now, get rid of the “/3“:

  • Start with:
x/3 = 3
  • If you multiply by 3 you can cancel out the divide by 3:
x/3 ×3 = 3 ×3
  • A little arithmetic (3/3 = 1 and 3×3 = 9) becomes:
x = 9

When you get more experienced:

When you get more experienced, you can solve it like this:

  • Start with:
x/3 + 2 = 5
  • Subtract 2:
x/3 = 3
  • Multiply by 3:
x = 9

 

Have a Try Yourself

Now practice on this Simple Algebra Worksheet and then check your answers from the choices.

Try to use the steps we have shown you here, rather than just guessing!

1.

Solve 3x = 9

Choose one:

6

3

12


2
.

Solve x/2 = 3

Choose one:

6

5

9

3.

Solve x/5 + 2 = 6

Choose one:

15

10

20


4
.

Solve 3x – 2 = 7

Choose one:

3

6

1


5
.

Solve 2(x + 3) = 10

Choose one:

3

2

5

Introduction to Algebra – Addition and Subtraction

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Algebra is great fun – you get to solve puzzles!

A Puzzle

What is the missing number?

OK, the answer is 6, right? Because 6 – 2 = 4. Easy stuff.

Well, in Algebra we don’t use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we would write:

The letter (in this case an x) just means “we don’t know this yet”, and is often called the unknown or the variable.

And when you solve it you write:

Why Use a Letter?

Because:
it is easier to write “x” than drawing empty boxes (and easier to say “x” than “the empty box”).
if there were several empty boxes (several “unknowns”) we can use a different letter for each one.

it doesn’t have to be x, it could be y or w … or any letter or symbol you like.

How to Solve

Algebra is just like a puzzle where you start with something like

“x-2 = 4” and you want to end up with something like “x = 6”.

But instead of saying “obviously x=6”, use this neat step-by-step approach:

  • Work out what to remove to get “x = …”
  • Remove it by doing the opposite (adding is the opposite of subtracting)
  • Do that to both sides

Here is an example:

We want to remove the “-2

To remove it, do the opposite, in this case add 2 to both sides:

Which is …

Solved!

Just remember this:

What you do to one side of the “=” you should also do to the other side!

Another Puzzle

Solve this one:  x + 5 = 12

  • Start with:
x + 5 = 12
  • What you are aiming for is an answer like “x = …”,
  • So, let us have a go at subtracting 5 from both sides:
x+5 -5 = 12 -5
  • A little arithmetic (5-5 = 0 and 12-5 = 7) becomes:
x+0 = 7
  • Which is just:
x = 7

Have a Try Yourself

Now practice on this Simple Algebra Worksheet and then check your answers from the choices.

Try to use the steps we have shown you here, rather than just guessing!

1.

Solve x + 2 = 6

Choose one:

6

4

8


2
.

Solve x – 3 = 8

Choose one:

9

11

13

3.

y – 8 = 7

Choose one:

1

-1

15


4
.

7 + y = 12

Choose one:

5

19

9


5
.

10 – y = 6

Choose one:

-4

4

16

Then read Introduction to Algebra – Multiplication and Division

Exponents

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Exponents are also called Powers or Indices. The exponent of a number says how many times to use the number in a multiplication.

Example: 53 = 5 × 5 × 5 = 125

In words: 53 could be called “5 to the third power”, “5 to the power 3” or simply “5 cubed”

Example: 24 = 2 × 2 × 2 × 2 = 16

In words: 24 could be called “2 to the fourth power” or “2 to the power 4” or simply “2 to the 4th”

Exponents make it easier to write and use many multiplications

Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

In general:

an tells you to multiply a by itself, so there are n of those a’s:

Negative Exponents

A negative exponent means how many times to divide one by the number.

Example: 8-1 = 1 ÷ 8 = 0.125

You can have many divides:

Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

5-3 could also be calculated like:   1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

In General

More Examples:

Negative Exponent Reciprocal of Positive Exponent Answer
4-2 = 1 / 42 = 1/16 = 0.0625
10-3 = 1 / 103 = 1/1,000 = 0.001
(-2)-3 = 1 / (-2)3 = 1/(-8) = -0.125

What if the Exponent is 1, or 0?


If the exponent is 1, then you just have the number itself (example 91 = 9)

If the exponent is 0, then you get 1 (example 90 = 1)
But what about 00 ? It could be either 1 or 0, and so people say it is “indeterminate”.

It All Makes Sense

My favorite method is to start with “1” and then multiply or divide as many times as the exponent says,

then you will get the right answer, for example:

Example: Powers of 5
52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.

Be Careful About Grouping

To avoid confusion, use parentheses () in cases like this:

(-2)2 = (-2) × (-2) = 4
22 = -(22) = – (2 × 2) = -4
(ab)2 = ab × ab
ab2 = a × (b)2 = a × b × b

 Your Turn

1.

What is the value of 34?

Choose one:

12

27

81


2
.

What is the value of 2-4?

Choose one:

0.125

-8

0.0625

3.

What is the value of 5-2?

Choose one:

0.04

-25

0.1


4
.

What is the value of (-3)5?

Choose one:

-125

243

-243


5
.

What is the value of (-2)-5?

Choose one:

-0.03125

0.03125

32

6.

What is the value of (-10)0?

Choose one:

zero

1

-10