Math 6 Honors Periods 1, 6 & 7 (Thursday)

Combined Operations 8-5

In order to solve an equation of the form

ax + b = c or ax –b = c or b – ax = c

where a, b, c are given numbers and x is the variable, we must use more than one transformation

Solve the equation 3n - 5 = 10 + 6

Simplify the numerical expression

3n - 5 = 10 + 6
3n – 5 = 16

add 5 to both sides

3n – 5 + 5 = 16 + 5
or
3n – 5 = 16
+ 5 = +5

3n = 21

divide both sides by 3 (or multiply each side by the reciprocal of 3)

3n/3 = 21/3

n = 7

General procedures for solving equations

Simplify each side of the equation
If there are still indicated additions or subtractions, use the inverse operation to undo them
If there are indicated multiplications or division involving the variable, use the inverse operations to undo them

The books says you must always perform the same operation on both sides of the equation. I say, “do to one side what you have done to the other side.”

Solve the equation
(3/2)n + 7 = 22

subtract 7 from both sides

(3/2)n + 7 - 7 = 22 - 7

(3/2)n =15


multiply both sides by 2/3, the reciprocal of 3/2

(2/3)(3/2)n = 15(2/3)

n = 10

Solve the equation
40 – (5/3)n = 15

add (5/3)n to both sides

40 – (5/3)n + (5/3)n = 15 + (5/3)n


40 = 15 + (5/3)n
subtract 15 from both sides

40 – 15 = 15-15 + (5/3)n

25 = (5/3)n multiply both sides by 3/5

(3/5)(25) = (5/3)n (3/5)

15 = n

Math 6 H Periods 1, 6 & 7 (Wednesday)

Equations: Decimals and Fractions 8-4


You can use transformations to solve equations which involve decimals or fractions

Solve 0.42 x = 1.05
Divide both sides by 0.42
.42x/.42 = 1.05/.42

now, do side bar and actually divide carefully and you will arrive at
x = 2.5

Solve: n/.15 = 92
multiply both sides by .15 to undo the division

(n/.15)(.15) = 92 (.15)
Again, do a sidebar for your calculations and you will arrive at

n = 13.80

How would we solve the following: (2/3)x = 6?

Let’s look at 2x = 6. What do we do?
We divide both sides by 2—or multiply both sides by the reciprocal of 2—which is ½
Remember the product of a number and its reciprocal is 1

Reminder: the ultimate objective is applying transformations to an equation is to obtain an equivalent equation in the form x = c
(where c is a constant.) Also remember that the understood (invivisble) coefficient of x in the equation x = c is 1. [Can you picture the poster in the front of the room?]
So to solve (2/3)x = 6 you would divide both sides by 2/3 but that is the same as multiplying by the reciprocal of 2/3, which is 3/2.

If an equation has the form

(a/b)(x) = c,
where both a and c are nonzero,
multiply both sides by b/a, the reciprocal of a/b
Solve (1/3)y = 18

(3/1)(1/3)y = 18(3/1)

y =18(3)
y = 54

Solve the equation: (6/7)n = 8

(7/6)(6/7)n = 8(7/6)

n = 8(7/6)

simplify first , then multiply
n = 28/3
n = 9 1/3
Let’s check

(6/7)n = 8
well, we said that n = 9 1/3 so substitute back, but change to 28/3 first
(6/7)(28/3) ?=? 8
[read ?=? as ‘does that equal?’]

Now really do a side bar with the left side of the equation to see what
(6/7)(28/3) really equals. Simplify before you multiply
2(4) = 8 so
8 = 8



Try:
1. (1/7)a = 13

2. b/8 = 16


3. 3.6d = 0.9


4. (3/8)f = 129

Math 6 H Periods 1, 6 & 7 (Tuesday)

Equations: Multiplication and Division 8-3

If an equation involves multiplication or division, the following transformations are used to solve the equation:

Transformation by multiplication: Multiply both sides of the equation by the same nonzero number.

Transformations by division: Divide both sides of the equation by the same nonzero number.

Remember: Do undo on one side what you would do undo the other!!

Our goal is to get the variable alone and to find an equivalent equation of the form
“n = a number”
Our goal is to arrive at the “world’s easiest equation”

Solve 3n = 24
Use the fact that multiplication and division are inverse operations
3n/3 = 24/3

n = 8

Or you could have use the reciprocal of 3-- which is 1/3 and multiplied both sides by 1/3
(1/3)(3n) = 24(1/3)
n = 8 and still arrived at the SAME solution

Solve 5x/5 = 53/5

x = 10 3/5

Solve n/4 = 7

(4)(n/4) = 7(4)

n = 28
How could you know for sure your answer is correct?

Substitute your solution into the ORIGINAL equation
Try:
1. 3r = 57

2. 714 = 7t

3. Solve A = bh for h

4. Solve P = 4s for s

5. Solve C = 2πr for r

Algebra Period 3 (Monday)

Direct Variation, Indirect Variation and Joint Variation: 12-5 TO 12-7


DIRECT VARIATION
As x increases, y also increases (graphs as a LINE)
or as x decreases, y also decreases
x and y go in the SAME DIRECTION
For example: y = 2x
the "2" is called the constant of variation and is "k" in the formula:
y = kx
To find k, you need one (x, y) coordinate to solve.
It's like solving for "m" in y = mx + b, but there is no b!
Example: y varies directly with x. When y = 8, x = 2. Find y when x = 3.
FORMULA FOR DIRECT VARIATION: y = kx
PLUG IN THE (x, y):
8 = k(2)
SOLVE FOR k:
k = 4
WRITE THE EQUATION:
y = 4x
PLUG IN VARIABLE GIVEN:
y = 4(3)
SOLVE FOR MISSING VARIABLE:
y = 12

INDIRECT VARIATION
As x increases, y decreases (graphs as a RATIONAL function...I'll show you it in class)
x and y go in OPPOSITE DIRECTIONS
For example: y = 2/x
the "2" is still called the constant of variation and is "k" in the formula:
y = k/x
To find k, you need one (x, y) coordinate to solve.
Example: y varies indirectly with x. When y = 8, x = 2. Find y when x = 3.
FORMULA FOR INDIRECT VARIATION:
y = k/x
PLUG IN THE (x, y):
8 = k/2
SOLVE FOR k:
k = 16
WRITE THE EQUATION:
y = 16/x
PLUG IN VARIABLE GIVEN:
y = 16/3
SOLVE FOR MISSING VARIABLE:
y = 5 1/3

JOINT VARIATION
TWO VARIABLES VARY WITH y AT THE SAME TIME!
As x AND z increase, y also increases (graphs as a LINEAR function)
The PRODUCT xz and y go in the SAME DIRECTION
For example: y = 2xz
the "2" is still called the constant of variation and is "k" in the formula:
y = kxz
To find k, you need one (x, y, z) coordinate to solve.
Example: y varies jointly with x and z. When y = 12, x = 2, and z = 3. Find y when x = 3 and z = 5
FORMULA FOR JOINT VARIATION:
y = kxz
PLUG IN THE (x, y, z):
12 = k(2)(3)
SOLVE FOR k:
k = 2
WRITE THE EQUATION:
y = 2xz
PLUG IN VARIABLE GIVEN:
y = 2(3)(5)
SOLVE FOR MISSING VARIABLE:
y = 30

Math 6 H Periods 1, 6 & 7 (Monday)

Equations: Addition and Subtraction 8-2



If the replacement set for an equation is the set of whole numbers, it is not practical to use substitution to solve the equation. Instead we transform or change the given equation into a simpler, equivalent equation. When we transform the given equation, our goal is to arrive at an equivalent equation of the form
variable = number

for example

n = 5

the number 5 is then, the solution of the original equation. The following transformations can be used to solve equation

Transformation by addition: add the same number to both sides
Transformation by subtraction: subtract the same number from both sides

solve x – 2 = 8

our goal is to find an equivalent equation of the form

x = a number

The left side of the given equation is x – 2. Recall that addition and subtraction are inverse operations. If we add 2 to both sides the left sides simplifies to x

x-2 = 8
x – 2 + 2 = 8 + 2 (We usually show the +2 right below each side of the equation)
x = 10

The solution is 10

Solve x + 6 = 17

Subtract 6 from both sides of the equation to get an equivalent equation of the form “ x = a number”

x + 6 = 17
x + 6 – 6 = 17 – 6 (Again, we usually show the -6 right below each side of the equation)
x = 11
the solution is 11

In equations involving a number of steps, it is a good idea to check your answer. This can be done easily by substituting the answer in the original equation.
What about the following

34 – x = 27

add x to both sides

34 – x + x = 27 + x
34 = 27 + x
subtract 27 from both sides

34 – 27 = 27 – 27 + x
7 = x

You may be able to solve some of the equations in the homework without pencil and paper. Nevertheless, it is important to show all the steps in your work and to make sure you can tell which transformation you are using in each step.

Which transformation was used to transform the first equation into the second

x – 3 = 5
x – 3 + 3 = 5 + 3


Remember when we stated the properties as well-- back in our 1st quarter!!