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

 

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