Direct variation describes the relationship between two quantities where one quantity varies as the other changes proportionally. It can be represented by the equation y = kx, where k is the constant of variation.
Some key points about direct variation:
- The graph of a direct variation will pass through the origin, as there is no y-intercept term.
- To determine if a relationship represents direct variation, calculate the constant of variation k from the data and check if it remains the same for different values.
- Direct variation can be used to find unknown values by setting up a table with the known values and using the direct variation equation y = kx.
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
You've seen that many quantities are related to each other. However, not all of them are directly related. Now you will explore quantities that vary inversely. In inverse variation, one quantity decreases as the other increases.
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
You've seen that many quantities are related to each other. However, not all of them are directly related. Now you will explore quantities that vary inversely. In inverse variation, one quantity decreases as the other increases.
All around us, some quantities are constant and others are variable.
For instance, the number of hours in a day is constant, but the number of hours in a daylight in a day is not.
now you will explore more closely certain types of relationships between variables.
This will help you in factoring sum and difference of two cubes.
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You already know relationships where one variable varies directly or inversely with another variable.
Now you will look at relationships where one variable varies directly with two or more other variables but does not vary inversely with any other variable.
Mathematics 9 Lesson 4-C: Joint and Combined VariationJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Joint and Combined Variations. It also discusses and explains the rules, concepts, steps and examples of Joint and Combined Variation
This learner's module discusses about the topic Variations. It also discusses the definition of Variation. It also discusses or explains the types of Variations. It also shows the examples of the Types of Variations.
All around us, some quantities are constant and others are variable.
For instance, the number of hours in a day is constant, but the number of hours in a daylight in a day is not.
now you will explore more closely certain types of relationships between variables.
This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
You already know relationships where one variable varies directly or inversely with another variable.
Now you will look at relationships where one variable varies directly with two or more other variables but does not vary inversely with any other variable.
Mathematics 9 Lesson 4-C: Joint and Combined VariationJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Joint and Combined Variations. It also discusses and explains the rules, concepts, steps and examples of Joint and Combined Variation
This learner's module discusses about the topic Variations. It also discusses the definition of Variation. It also discusses or explains the types of Variations. It also shows the examples of the Types of Variations.
May this presentation could help you, the pictures here is not mine I get it from YouTube videos, I upload this ppt because
most the ppt here help me a lot in my teaching mathematics. This topics is different kinds variation direct variation, inverse variation, joint variation and combined variation.
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He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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2. Definition:
Y varies directly as x means that y = kx
where k is the constant of variation.
(see any similarities to y = mx + b?)
Another way of writing this is k =
In other words:
* the constant of variation (k) in a direct
variation is the constant (unchanged) ratio of two
variable quantities.
y
x
3. Examples of Direct Variation:
X Y
6 12
7 14
8 16
Note: X increases,
6 , 7 , 8
And Y increases.
12, 14, 16
What is the constant of variation of the table above?
Since y = kx we can say Therefore:
12/6=k or k = 2 14/7=k or k = 2
16/8=k or k =2 Note k stays constant.
y = 2x is the
equation!
y
k
x
=
4. X Y
10 30
5 15
3 9
Note: X decreases,
10, 5, 3
And Y decreases.
30, 15, 9
What is the constant of variation of the table above?
Since y = kx we can say Therefore:
30/10=k or k = 3 15/5=k or k = 3
9/3=k or k =3 Note k stays constant.
y = 3x is the
equation!
y
k
x
=
Examples of Direct Variation:
5. X Y
-4 -1
-16 -4
-40 -10
Note: X decreases,
-4, -16, -40
And Y decreases.
-1,-4,-10
What is the constant of variation of the table above?
Since y = kx we can say Therefore:
-1/-4=k or k = ¼ -4/-16=k or k = ¼
-10/-40=k or k = ¼ Note k stays constant.
y = ¼ x is the
equation!
y
k
x
=
Examples of Direct Variation:
6. What is the constant of variation for the
following direct variation?
X Y
4 -8
8 -16
-6 12
3 -6
Answer
Now 2
-2
-½
½
0% 0%0%0%
1. 2
2. -2
3. -½
4. ½
7. Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
X Y
4 6
8 12
12 18
18 27
Yes!
k = 6/4 or 3/2
Equation?
y = 3/2 x
8. X Y
10 25
6 15
4 10
2 5
Yes!
k = 25/10 or 5/2
k = 10/4 or 5/2
Equation?
y = 5/2 x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
9. X Y
15 5
3 26
1 75
2 150
No!
The k values are
different!
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
10. Which of the following is a direct variation?
A
B
C
D
0% 0%0%0%
1. A
2. B
3. C
4. D
Answer
Now
12. Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 28 when
x=7, Find x when y = 52. HOW???
2 step process
X Y
7 28
? 52
1. Find the constant variation
k = y/x or k = 28/7 = 4
k=4
2. Use y = kx. Find the unknown (x).
52= 4x or 52/4 = x
x= 13
Therefore:
X =13 when Y=52
13. Given that y varies directly with x, and y = 3 when x=9,
Find y when x = 40.5. HOW???
2 step process X Y
9 3
40.5 ?
1. Find the constant variation.
k = y/x or k = 3/9 = 1/3
K = 1/3
2. Use y = kx. Find the unknown (x).
y= (1/3)40.5
y= 13.5
Therefore:
X =40.5 when
Y=13.5
Using Direct Variation to find unknowns (y = kx)
14. Given that y varies directly with x, and y = 6 when x=-5,
Find y when x = -8. HOW???
2 step process
X Y
-5 6
-8 ?
1. Find the constant variation.
k = y/x or k = 6/-5 = -1.2
k = -1.2
2. Use y = kx. Find the unknown (x).
y= -1.2(-8)
x= 9.6
Therefore:
X =-8 when Y=9.6
Using Direct Variation to find unknowns (y = kx)
15. Using Direct Variation to solve word problems
Problem:
A car uses 8 gallons of
gasoline to travel 290
miles. How much
gasoline will the car use
to travel 400 miles?
Step One: Find points in table
X (gas) Y (miles)
8 290
? 400
Step Two: Find the constant
variation and equation:
k = y/x or k = 290/8 or 36.25
y = 36.25 x
Step Three: Use the equation
to find the unknown.
400 =36.25x
400 =36.25x
36.25 36.25
or x = 11.03
16. Using Direct Variation to solve word problems
Problem:
Julio wages vary
directly as the number
of hours that he works.
If his wages for 5 hours
are $29.75, how much
will they be for 30 hours
Step One: Find points in table.
X(hours) Y(wages)
5 29.75
30 ?
Step Two: Find the constant
variation.
k = y/x or k = 29.75/5 = 5.95
Step Three: Use the equation
to find the unknown. y=kx
y=5.95(30) or Y=178.50
17. Direct Variation and its graphDirect Variation and its graph
y = mx +b,
m = slope and b = y-intercept
With direction variation the equation
is y = kx
Note: m = k or the constant and b = 0 therefore the graph will
always go through…