Algebra Basics

Substitution

What is substitution?

Substitution is where we replace letters in a formula with their values
This allows you to find one other value that is in the formula

How do we substitute numbers into a formula?

Write down the formula, if not clearly stated in question
Substitute the numbers given, using brackets around negative numbers, (-3), (-5) etc
Simplify any calculations if you can
Rearrange the formula if necessary (it is usually easier to substitute first)
Work out the calculation (use a calculator if allowed)

Are there any common formulae to be aware of?

Formulae for accelerating objects are often used
- $v = u + a t$
- $v^{2} = u^{2} + 2 a s$
- $s = u t + \frac{1}{2} a t^{2}$
The letters mean the following:
- t stands for the amount of time something accelerates for (in seconds)
- u stands for its initial speed (in m/s) - the speed at the beginning
- v stands for its final speed (in m/s) - the speed after t seconds
- a stands for its acceleration (in m/s²) during in that time
- s stands for the distance covered in t seconds
You do not need to memorise these formulae, but you should know how to substitute numbers into them
- You should also understand the difference between initial speed (u) and final speed (v)

Worked example

(a)

Find the value of the expression

2 x (x + 3 y)

when

x = 2

and

y = - 4

Substitute the numbers given.
Use brackets () around negative numbers that you have substituted so that you don't forget about them.
It is a good idea to show every step of working to make sure that you are following the order of operations correctly.

$\begin{array}{rcl} 2 \times 2 \times (2 + 3 \times (- 4)) \\ = & 2 \times 2 \times (2 - 12) \\ = & 2 \times 2 \times (- 10) \\ = & 4 \times - 10 \end{array}$

$- 40$

(b)

The formula

P = 2 l + 2 w

is used to find the perimeter, P, of a rectangle of length

l

and width

w

Given that the rectangle has a perimeter of 20 cm and a width of 4 cm, find its length.

Substitute the values you are given into the formula.

$20 = 2 \times l + 2 \times 4$

Simplify.

$20 = 2 l + 8$

Subtract 8 from both sides.

$12 = 2 l$

Divide both sides by 2.

$l = 6 cm$

Collecting Like Terms

How do we collect like terms?

TERMS are separated by + or –
- The sign belongs to the coefficient of the term after the symbol
- If there is no symbol in front of the first term then this is a positive term
  - 2x - 3y means +2 x's and -3 y's
“LIKE” terms must have exactly the same LETTERS AND POWERS (the COEFFICIENT can be different)
- Examples of like terms:
  - 2x and 3x
  - 2x² and 3x²
  - 2xy and 3xy
  - 4(x + y) and 5(x + y)
- Examples that are NOT like terms
  - 2x and 3y (different letters)
  - 2x² and 3x⁴(different powers)
  - 2xy and 3xyz (different letters)
  - 4(x + y) and 5(x + y)² (different powers)
- Remember multiplication can be done in any order
  - xy and yx are like terms
Add the COEFFICIENTS of like terms
- If the answer is a positive answer then put "+" in front if there are other terms before it
  - x - 2y + 5y = x + 3y
- If the answer is a negative number then put "-" in front
  - x - 5y + 2y = x - 3y

Exam Tip

A “coefficient” answers the question “how many?”

For example:

the coefficient of x in 2x² – 5x + 2 is -5

and:

the coefficient of x in ax² + bx + c is b

Worked example

Simplify $x^{2} - 3 x y + 2 x^{2} + 4 x - 2 x y - x + 7$ .

Reorder the terms so that the $x^{2}$ 's are together, as are the $x y$ 's, $x$ 's and the constants.
Make sure that you keep the same sign in front of them when you reorder them.

$x^{2} + 2 x^{2} - 3 x y - 2 x y + 4 x - x + 7$

Add the coefficients of the 'like' terms.

$3 x^{2} - 5 x y + 3 x + 7$