Download 2º ESO BIL Dpto. de Matemáticas- I.E.S. Montes Orientales (Iznalloz

2º ESO BIL Dpto. de Matemáticas- I.E.S. Montes Orientales (Iznalloz)- Nieves Fiestas Carmona. Curso 2011/2012
UNIT 6. EQUATIONS
EQUATIONS AND THEIR COMPONENTS
What is an equation? Components
An equation is an expression stating the equality of tow
algebraic expressions. This equality exists only for certain
values of the letters.
Ex: x + 6 = 2x + 1, is true for x = 5 ( 5 + 6 = 2 · 5 + 1 )
We call each side of the equal sign, expression, the letters
are called unknowns, the addends are called terms and the
values for which the equation is true are called solutions.
Ex: x + 6 = 2x + 1
First expression: x + 6. Second expression: 2x + 1.
Unknown: x. Terms: x, 6, 2x, 1. Solution: x = 5.
BASIC EQUATION-SOLVING TECHNIQUES
Adding or Subtracting from both expressions
If the same amount is added or subtracted from both
expressions in an equation, the result is another, equivalent
equation (with the same solution).
Rule: A number that is added /subtracted in one expression
can be eliminated from that expression if it is subtracted
/added from the other expression.
Ex 1: x + 2 = 5 -( x + 2 – 2 = 5 – 2) - x = 5 – 2 - x = 3
Ex 2: x - 1 = 3 -( x - 1 + 1 = 3 + 1) - x = 3 + 1 - x = 4
Dividing or Multiplying both expressions
If both expressions in an equation are divided or multiplied
by the same number, the result is another, equivalent
equation.
Rule: A multiplication / division operation can be eliminated
from one expression if it is added as a division /
multiplication operation to the other expression.
Ex 1: 2x = 6 -( 2x / 2 = 6 / 2) - x = 6 / 2 - x = 3
Ex 2:
x
 4 
3
x

  3  (4)  3  - x = - 12
3

FIRST DEGREE EQUATIONS
A first-degree equation is an equation than can be reduced
to the form ax=b. It has a single solution of x = b/a.
Solving equations
To solve a first degree equation with one unknown, you
reduce the expressions and move the terms until the
unknown is isolated.
Ex: 6x - 4 = 3x + 6
6x – 3x = 6 + 4 First, we move the terms
3x = 10
Second, we reduce the terms adding
x=
10
3
Third, we work out the unknown
SOLVING PROBLEMS
Equations as tools
b)
Equations can help you to solve problems. In each of the
sample equations, study the steps that must be followed.
Ex: A number and the same number plus one add up to 43.
What are the two numbers that add up to 43?
a)
Identify the known and unknown numbers:
A number
The number plus one
The sum of both
n
n+1
43
Find the relationship between the variables.
A number + The same number plus one = 43
Equation: n
c)
+
(n + 1)
= 43
Solve the equation.
n + n + 1 = 43  2n = 43 – 1  2n = 42  n = 21
d) Write the solution:
n = 21 ; n + 1 = 21 + 1 = 22
The numbers are 21 and 22