B asic Algebra

Some Rules

Order of operations:

1. Parentheses - type of brackets makes no difference.

2. Exponents.

3. Multiplying and dividing from left to right.

4. Addition and subtraction from left to right.

Multiplication forms

 9 * x  9x  9(x)

Division forms

9÷x 9/x

Zero times any number is zero and 1 times any number is the number.  Examples: 

x (0) = 0 (0) x = 0 x (1)= x 1 * x = x

Two sides of an equation are always equal, so as long as you multiply or divide each side by the same amount, it will still be equal.  Examples:

a = b, ac = bc (a / c) = (b / c)

Powers

You can only add or subtract the same letter raised to the same power - so x² + x² can be added but not x² + x³

x² + x² = 2 x²

x² x x² = x⁴ (add the indices for multiplication)

x⁶ ÷ x² = x⁴ (subtract the indices for division)

Terms: 3 x 3 is the coefficient x is the variable

Examples

1. 2x + 5y − x + 3a

We can only combine the x terms.

→ (2x − x) + 3a + 5y

→ = 3a + x + 5y

(Variables are usually written in alphabetical order.)

2. -2[-3(x − 2y) + 4y]

Let’s expand the round bracket part first

→ -3(x − 2y)

→ -3x − (-3)(2y)

→ -3x + 6y (note the double negative becomes a positive)

Now let’s put the -2 outside back and work it in

→ -2[-3x + 6y + 4y]

→ -2[-3x + 10y] (add the ys)

→ = 6x − 20y

Practice – your turn!

1. 4x − -5y + 10x + 3a+ -2x + 3y - a

2. - [7(a − 2b) − 4b]

3. y ² + y²

4. y ⁴ + y⁵ + 4 y⁵

5. a ⁸ ÷ a³

6. x²(x⁴ + 6)

7. (2x + 4)(x² – 6 − x)

8. 10x[- 4 + 2(x − y)] + 7x

Answers

1. 4x − -5y + 10x + 3a+ -2x + 3y - a

(4x+ 10x + -2x) – (-5y + 3y) + (3a- a) (group letters that are the same)

12 x - -2y+ 2a (double negative equals a positive)

2a + 12 x + 2y (put in alphabetical order)

2. - [7(a − 2b) − 4b]

- [7a − 14b − 4b]

- [7a − 18b]

= - 7a + 18b

3. y ² + y² = 2 y²

4. y ⁴ + y⁵ + 4 y⁵ = y⁴ + 5y⁵

5. a ⁸ ÷ a³ = a⁵

6. x²(x⁴ + 6)

= x⁶ + 6x²

7. Take the 2 terms of the first bracket and multiply both of them by the second bracket.

(2x + 4)(x² – 6 − x)

= (2x)(x² – 6 − x) + (4)(x² – 6 − x)

= (2x³ − 12x− 2x²) + (4x² −24 − 4x)

= 2x³ + 2x² − 16x – 24 (combine any like terms)

8. 10x[- 4 + 2(x − y)] + 7x

= 10x[- 4 + 2x − 2y] + 7x (expand the inside set of brackets first)

= -40x + 20x² − 20xy + 7x (expand the next set of brackets)

= -33x + 20x² − 20xy

Fractions

Terms: a is the numerator, b is the denominator. They are both integers. b cannot be 0.

To multiply fractions – multiply the tops together and then the bottoms together.

x = =

To divide – simply invert the fraction after the division sign and then multiply as above.

÷ = x =

To add or subtract, they must have the same denominator – and then remember that the denominator stays the same – just add or subtract the top numbers.

+ = – =

If the denominators aren’t the same, multiply each numerator by the opposite denominator then multiply the denominators together to get the new denominator.

Reduce fractions by dividing the same number into the top and bottom – in the example below, both can be divided by 4, giving the same number (but in a different form).

+ = + = = =

Practice – your turn!

1. x =

2. ÷ =

3. + 1 =

4. - =

Answers

1. x = (Then reduce the fraction further by dividing 2 into each number)

=

2. ÷ =

= x (Remember that to divide we invert the second fraction and then multiply)

= (Then divide both by 3)

=

The numerator is greater than the denominator so divide the bottom into the top and that gives you the whole number and then the remainder is the new numerator.

=1

3. + 1 =

First assimilate the whole number on the left into the fraction by multiplying it by the denominator and then adding it to the numerator

(because the dominators are the same you can just add the whole numbers and then the fractions, but it’s a good idea to get into the habit of putting whole numbers into a fraction in case the denominators aren’t the same).

= +

=

= 4

4. - =

= -

= =

=

Further Resources

http://www.intmath.com/basic-algebra/basic-algebra-intro.php

http://library.thinkquest.org/20991/prealg/eq.html

http://www.mathleague.com/help/algebra/algebra.htm

Google terms or questions and always check for videos if you’d prefer to watch someone do it instead of reading.

www.youtube.com

www.ehow.com