OHP worksheets

Equations
Table of contents
1.
What equations are used for
2.
Idea and definition
3.
Some concepts useful in dealing with equations
4.
How to get the solutions of an equation
5.
Rules for operating on equations
6.
Strategies to solve equations
7.
Linear equations

WHAT EQUATIONS ARE USED FOR
If you need to solve a problem you can separate information into two
“equal” groups

EQUATIONS
IDEA:
An equation represents a BALANCE, such as the one shown below, which is kept balanced by two expressions
DEFINITION
An EQUATION is an equality between two mathematical expressions, in which there is (at least) one unknown (i.e. a variable that stands for an unknown value).
The unknown values that make the equality true must be looked for.
Examples: 10x = 0
9x 2 = 9
2x+1= -3x+15
7x+ 5x 3 = 9y + 3
LEFT SIDE = RIGHT SIDE

SOME CONCEPTS USEFUL IN DEALING WITH
EQUATIONS
SUBSTITUTION : in an expression
To substitute [a value] for [a variable] = to put [ a value] in the place of [a variable]
To replace [ a variable] with [a value]
SOLUTION : each value that makes the equation a true equality by being substituted for the unknown
[the left side comes out equal to the right side]
VERIFICATION : CHECK OF THE SOLUTION
You must substitute your solution in the original equation and you should obtain a true statement
…… some examples

IDENTITY : equality between two expressions which is true whatever values are substituted for the variables occurring in it.
Examples: 5x + 4 – 3 = 7x – 2x + 1 a 2 – b 2 = (a – b)(a + b)
SOLUTION SET OF AN EQUATION : the set consisting of all its solutions
…… some examples
EQUIVALENT EQUATIONS :
Two equations are EQUIVALENT if they have the same solution set
…… some examples

HOW
?
OPTIONS

By inspection (intelligent guessing)

By a formal method (using rules)
TO SOLVE an equation = to find all its solutions
SOLVING PROCESS
In solving equations by a formal method, your task is to write a sequence of EQUIVALENT EQUATIONS, which have to be simpler and simpler, until you get the unknown you are solving for on its own.
Starting from the equation you need to solve, each following equation is obtained from the previous one by using some rules.

RULES FOR OPERATING ON EQUATIONS
BASIC PRINCIPLE
The same operation must be performed on both sides of the equation
…… OTHERWISE THE BALANCE
WILL GET UNBALANCED

ADDITION / SUBTRACTION PROPERTY
THE TWO SIDES OF AN EQUATION REMAIN EQUAL IF
ANY NUMBER IS ADDED TO / SUBTRACTED FROM BOTH
SIDES
A = B
A + C = B + C A – C = B – C
…… some examples
MULTIPLICATION / DIVISION PROPERTY
THE TWO SIDES OF AN EQUATION REMAIN EQUAL IF
BOTH SIDES ARE MULTIPLIED / DIVIDED BY ANY
NUMBER
A = B
A ∙ C = B ∙ C A : C = B : C
…… some examples

TO SOLVE EQUATIONS
1.
In solving equations, the properties worded in the previous transparency are used to write simpler and simpler equations, until the unknown we are solving for is got alone.
2.
To do this, you need to:
UNDO WHAT THE EQUATION IS DOING
i.e.
PERFORM THE INVERSE OPERATION:
ADDITION ← → SUBTRACTION
MULTIPLICATION ← → DIVISION

LINEAR EQUATIONS
LINEAR EQUATION = an equation that is equivalent to one which has the form A ∙ x = B , where A and B are constants and x is the unknown.
In particular, the unknown never occurs:

In denominators

In divisors

Under root signs
Sometimes it can occur with an exponent greater than 1
HOW MANY SOLUTIONS?
GENERALLY: JUST ONE S =

B/A

SOMETIMES NONE S =

SOMETIMES EVERY NUMBER S = R