Transcript PPT for Section 2.2 and 2.3

WEEK 2
SECTION 2.2 AND SECTION
2.3
SRABASTI DUTTA
REVIEW OF PROPERTIES FROM WEEK 1 -- ADDITION
•Addition property of 0: adding zero to any number leaves that number unchanged.
Thus, 13 + 0 = 13
•Commutative property of addition: changing the order of numbers in an addition
does not affect the result: 26 + 37 = 37 + 26 = 63
•Associative property of addition: changing the grouping of numbers in an addition
does not affect the result: (2 + 3) + 5 = 2 + (3 + 5) = 10
•Distributive Property of addition and multiplication: 2(3 + 5) = 2*3 + 2*5 = 6 + 10 =
16; it is known as the distributive property because we are distributing 2 to 3 and 5
(multiplying 3 and 5 by 2).
REVIEW OF PROPERTIES FROM WEEK 1 -- MULTIPLICATION
1.
2.
3.
4.
5.
Distributive Property: the sum of two numbers times a third number is equal
to the sum of each number multiplied to the third number: 2(3 + 4) = 2*3 +
2* 4 = 6 + 8 = 14
Multiplication Property of zero: any number multiplied to zero gives us a
zero: 2 * 0 = 0, 7 * 0 = 0, 10 * 0 = 0
Multiplication property of 1: any number multiplied to one gives us back the
number: 2 * 1 = 2, 4 * 1= 4
Commutative property: the order of the numbers in the product does not
affect the result. Thus, 2*3 = 3*2 = 6
Associative property: Grouping does not affect the result: (2*3)*1 = 2(3*1)
=6
SIMPLIFYING EXPRESSIONS WITH VARIABLES
The first step in solving any equation is simplifying the given expression.
Such simplifications can be done by applying associative, distributive and
commutative properties of addition and multiplication.
Example 1: 3(2x) can be written as (3*2)x by applying the associative property
of multiplication. Thus, 3(2x) = (3*2)x = 6x is the simpler form.
Example 2: -4(8x) = (-4*8)x = -32x, by associative property.
Example 3: 2(x + 1) = 2x + 2, by the distributive property – distributing 2 to
both x and 1
SIMPLIFYING EXPRESSION WITH VARIABLES CONTINUES
Example 4: Simplify 4(x – 1) + 2
Step 1: Distribute 4 to x and -1 to get 4x – 4 + 2
Step 2: Combine -4 and 2 to get 4x +(-4 + 2) = 4x + (-2) = 4x – 2
Example 5: Simplify -2(3x – 5y) + 9
Step 1: Rewrite the expression as -2(3x + (-5y)) + 9
Step 2: Distribute -2 to get (-2*3)x + (-2*-5)y + 9 = -6x + 10y + 9
Example 6: Simplify 18x + 3x
Step 1: 18x + 3x = (18 + 3)x, by distributive property
Step 2: Add 18 + 3 to get 21x as the solution.
Example 7: Simplify 20x – 5x
Step 1: 20x – 5x = (20 – 5) x, by distributive property
Step 2: Subtract to get 15x as the solution
SIMILAR TERMS AND SIMPLIFICATION
Given 2x + 4x, 2x and 4x are known as similar terms because the variable
terms are same.
Similarly in 2x + 3 + 9, 3 and 9 are similar terms both of them are some
constant numbers and do not involve any variables.
Using the concept of similar terms, we can simplify more complicated
equations like 2x + 4x + 3 + 9.
Step 1: Combine (group together) the like terms (2x + 4x) + (3 + 9)
Step 2: Simplify each grouped terms. Thus, 2x + 4x = (2 + 4)x =
6x. And, 3 + 9 = 12.
Step 3: Write down the final solution as 6x + 12
SIMILAR TERMS AND SIMPLIFICATION -- EXAMPLES
Simplify 6x – 3 + 4x + 8
Step 1 Grouping: (6x + 4x) + (-3 + 8)
Step 2 Simplify each group: 6x + 4x = (6 + 4)x = 10x and -3 + 8 =
5
Step 3 Write down the answer: 10x + 5
Simplify 3(2x + 1) + 5(3x – 6)
Step 1: Distribute: 3*2x + 3*1 + 5*3x + 5*-6 = 6x + 3 + 15x – 30
Step 2: Combine like terms (6x + 15x) + (3 – 30)
Step 3: Simplify each group (6 + 15)x + (3 + -30) = 21x + -27 =
21x – 27
SOLUTIONS OF EQUATIONS
A solution for an equation is a number that when used in place of the variable
makes the equation a true statement.
Thus, for 2x + 3 = 7, the solution is x = 2 because:
Step 1: substitute x = 2: 2(2) + 3 = 7
Step 2: compute: 4 + 3 = 7
Step 3: simplify: 7 = 7
Step 4: reach conclusion: because 7 = 7 is a true statement, thus
x = 2 is the solution for 2x + 3 = 7
We found the solution through trial-and-error method. Next we
will learn how to find the solution by actually solving an
equation and not using trial and error method.
RULES TO REMEMBER AND FOLLOW FOR SOLVING
EQUATIONS
The following rules/steps are usually applied in solving equations. Thus, you should
really try to remember them:
1. Apply the distributive, associative, commutative properties of addition and
multiplication to simplify expressions.
2. Apply the addition/subtraction property of equality
3. Combine like terms
4. In an equation, all the terms involving variables should always be on one side
of the equation; all the constant numbers should be on the other side of the
equation.
RECIPROCAL
Reciprocal of a number is 1 divided by that number. Thus:
Example 1: Reciprocal of 2 is ½
Example 2: Reciprocal of 8 is 1/8
Example 3: Reciprocal of 1 is 1/1 which is nothing but 1
Example 4: Reciprocal of -4 is 1/-4 which is written as -1/4
Example 5: Reciprocal of -10 is 1/-10 or -1/10
NOTE: There is no reciprocal of 0 because 1/0 is not acceptable in
mathematics.
Reciprocal of -1 is 1/-1 which is nothing but -1
SOLVING EQUATIONS -- EXAMPLES
Solve 4x – 3 - 3x = 5 – 10
Step 1 combine like terms (4x – 3x) – 3 = 5 – 10
Step 2 simplify: (4-3)x – 3 = -5 x – 3 = -5
Step 3 add positive 3 on both sides: x – 3 + 3 = -5 + 3
Step 4 simplify by cancelling the positive and negative 3 to get: x = -2 and that is
our solution (we added positive 3 because that is the opposite of negative 3. So,
to get rid off any number, we always do the opposite operation. And the
operation needs to be done on both sides of the equation).
Check (optional): You can always check if you have the correct
solution or not. Substitute your solution in the given equation
and see if you get a correct statement:
4(-2) – 3 – 3(-2) = 5 – 10  -8 – 3 + 6 = -5  -11 + 6 = -5  -5 =
5 and a true statement
EXAMPLE CONTINUES
Solve 5x + 9 = 4x + 3
Step 1: Add negative 9 to both sides of the equation and
simplify: 5x + 9 + (-9) = 4x + 3 + (-9)  5x + 9 – 9 = 4x +(3 -9)  5x = 4x – 6
Step 2: Since there is a positive 4x on the right side of the
equation, we need to bring it back to the left side of the
equation because we need to have all the variable terms on one
side and numbers on the other. So, add negative 4x on both
sides of the equation and simplify: 5x + (-4x) = 4x – 6 + (-4x) 
5x – 4x = 4x – 6 – 4x  (5 -4)x = (4x – 4x ) – 6  x = 0 – 6  x =
6 and that is our solution.
EXAMPLE
Solve 5x – 8x + 3 = 4 – 10
Step 1: (5 – 8)x + 3 = 4 + (-10)
-3x + 3 = -6
Step 2: Add negative 3 to both sides to get -3x + 3 + (-3) = -6 + (
3)
-3x + (3 – 3) = -9
-3x = -9
Step 3: Multiply both sides by the reciprocal of -3 which is -1/3:
3 x 
1
1
 9 
3
3
Step 4: Simplify by cancelling out the 3’s to get: x = 3 as the
solution
EXAMPLE
Solve 6(3x – 6) + 10 = 5(2x + 8) – 10
Step 1 Distribute: 6*3x + 6*-6 +10 = 5*2x + 5*8 – 10
Step 2 Simplify: 18x + (-36) + 10 = 10x + 40 – 10
18x + (-36 + 10) = 10x + 30
18x -26 = 10x + 30
Step 3 Add positive 26 to both sides: 18x – 26 + 26 = 10x + 30 +
26 18x = 10x + 56
Step 4 Add negative 10x from both sides: 18x + (-10x) = 10x + 56
+ (-10x)
Step 5 Simplify: (18 – 10)x = (10 – 10)x + 56 8x = 56
Step 6 Multiply the reciprocal of 8 to both sides: 8 x  1  56  1
8
Step 7 Simplify by cancelling the 8’s to get
x = 7 and that is our solution
8
1
1
8 x   56 
8
8
EXAMPLE WITH DECIMAL
Your book is asking you to first multiply either by 10, 100, etc to get rid off
the decimal. It can become confusing. So, just follow the rules that we
discussed in the previous slides.
Example: Solve 0.5x + 3.4 = 0.2x + 12.4
Step 1: Subtract 3.4 from both sides: 0.5x + 3.4 – 3.4 = 0.2x + 12.4 – 3.4
Step 2: Cancel the positive and negative 3.4 from the left: 0.5x + 3.4 – 3.4 =
0.2x + 12.4 – 3.4 to get: 0.5x = 0.2x + 9
Step 3: Subtract 0.2x from both sides: 0.5x – 0.2x = 0.2x + 9 – 0.2x
Step 4: Simplify 0.3x = 9
0.3x
9

Step 5: Divide both sides by 0.3x:
0.3
0.3
Step 6: Simplify: x = 9/0.3 = 30 (using calculator)