MATH 9 VARIATIONS

DIRECT

JOINT

INVERSE

COMBINED

VARIATIONS

Q2 WEEK 1

assignment

DIRECT VARIATIONS

an increase in x causes an increase in y as well, Similarly, a decrease in x cause a decrease in y.

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DIRECT

VARIATION

PRACTICE

EXAMPLES

DEFINITION

DIRECT VARIATIONS

A bicycle is traveling 10 kilometers per hour (kph). In one hour it goes 10 kilometers (km). In two hours, it goes 20 km. In 3 hours, it goes 30 km and so on. Using the number of hours as the first number and the number of kilometers Traveled as the second number: (1,10), (2,20), (3,30), (4,40) and so on. Note that as the first number gets larger, so does the second. Note also, that the ratio of distance to time for each of these ordered pairs is a constant, or 10.

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DIRECT VARIATIONS

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DIRECT VARIATIONS

There is direct variation whenever a situation produces pairs of numbers in which their ratio is constant. Hence, the distance traveled by the bicycle varies directly as the time of travel. (10 is a constant) or

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Illustrative Example:

1. If y varies directly as x and that y=32 when x=4. Find the variation constant and the equation of variation.

ANSWER

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EXAMPLE 3

EXAMPLE 2

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PRACTICE EXERCISE

A. Write an equation for the following statements1. The fare F of a passenger varies directly as the distance d of his destination. 2. the cost C of a fish varies directly as its weight w in kilograms. 3. An employee’s salary S varies directly as the number of days d he has worked . 4. The distance D travelled by a car varies directly as its speed s. 5. The cost C of electricity varies directly as the number of kilowatts-hour consumption I.

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B. Find an equation where y varies directly as x.

1. y = 28 when x = 7

B. Find an equation where y varies directly as x.

2. y = 30 when x = 8

INVERSE VARIATIONS

THE STATEMENT, “ y IS INVERSELY PROPORTIONAL TO x ,” TRANSLATES TO y=k/x , WHERE IS THE PROPORTIONALITY CONSTANT OR CONSTANT OF VARIATION.

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INVERSE

VARIATION

PRACTICE

EXAMPLES

DEFINITION

INVERSE VARIATIONS

A car is traveling a distance of 10 km at a speed of 10 km/hr, and it will take one hour to finish the trip. At 30 km/hr, it will take 1/3 hr to finish the trip and so on. This determines a set of pairs of numbers, all having the same product: (10,1), (20,1/2 ), (30,1/3 ), (40,1/4 ) and so on . In table form the relation between the speed and the time of traveling is shown: Note that as the first number gets larger, the second number gets smaller.

INVERSE VARIATIONS

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Illustrative Example:

Find the equation and solve for k: y varies inversely as x and y = 6 when x = 18.

ANSWER

EXAMPLE 3

EXAMPLE 2

PRACTICE EXERCISE

A. Write an equation for the following statements1. The number of pizza slices p varies inversely as the number of persons n sharing a whole pizza. 2.The number of pechay plants n in in a row varies inversely as the space s between them. 3. The number of person n needed to do a job varies inversely as the number of days d to finish the job. 4.The rate r at which a person types a certain manuscript varies inversely as the time t spent in typing.

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B. Find an equation where y varies inversely as x.

1. y = 25 when x = 3

B. Find an equation where y varies inversely as x.

2. y = 7 when x = 10

JOINT VARIATIONS

The value of y varies directly to two or more quantities.

JOINT

VARIATION

PRACTICE

EXAMPLES

DEFINITION

JOINT VARIATIONS

Some physical relationships, as in area or volume, may involve three or more variables simultaneously. Consider the area of a rectangle which is obtained from the formula: A=lw where l is the length, w is the width of the rectangle. The table shows the area in square centimetres for different values of the length and the base.

Observe that A increases as either l or w increase or both. Then it is said that the area of a rectangle varies jointly as the length and the width.

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JOINT VARIATIONS

Consider the area of a triangle, which is obtained from the formula: a = (1/2) AB Where is the base and is the altitude of the triangle. The table shows the area in square centimetres for different values of the base and altitude, both being in centimetres.

b a

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Observe that A increases as either or increase or both. We say that the area of a triangle varies jointly as the base and the altitude.

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Illustrative Example:

Find an equation of variation where a varies jointly as b and c, and when a=36 , b=3 and c=4 .

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EXAMPLE 3

EXAMPLE 2

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PRACTICE EXERCISE

A. Write an equation for the following statements1. P varies jointly as q and r. 2. V varies jointly as l,w, and h. 3. The area A of a parallelogram varies jointly as the base b and altitude h. 4. The volume of a cylinder V varies jointly as its height h and the square of the radius r. 5. The heat H produced by an electric lamp varies jointly as the resistance R and the square of the current c.

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The area A of a triangle varies jointly as the base b and the altitude a of the triangle. If A=65cm^2 when b = 10cm and a = 13cm, find the area of a triangle whose base is 8cm and altitude is 11cm.

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The volume (V) of a prism on a square base varies jointly as the height (h) and the square of a side (s) of the base of the prism. If the volume is 81cm^3 when a side of the base is 4cm and the height is 6cm, write the equation of the relation.

ANSWER

COMBINED VARIATIONS

The value of 𝑦 varies directly to some quantities and varies inversely to some other quantities.

COMBINED

VARIATION

PRACTICE

EXAMPLES

DEFINITION

COMBINED VARIATIONS

Combined variation is another physical relationship among variables. This is the kind of variation that involves both the direct and inverse variations.

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Illustrative Example:

1. If z varies directly as x and inversely as y, and z = 9 when x = 6 and y = 2, find z when x = 8 and y = 12.

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EXAMPLE 3

EXAMPLE 2

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PRACTICE EXERCISE

A. Translate each statement into a mathematical statement. Use k as the constant of variation.1. T varies directly as a and inversely as b. 2. Y varies directly as x and inversely as the square of z. 3. P varies directly as the square of x and inversely as s 4. The time t required to travel is directly proportional to the temperature T and inversely proportional to the pressure P. 5. The pressure P of a gas varies directly as its temperature t and inversely as its volume V.

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1. If r varies directly as s and inversely as the square of u, then r = 2 when s = 18 and u = 2. Find r when u = 3 and s = 27.

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2. If r varies directly as s and inversely as the square of u, then r = 2 when s = 18 and u = 2. Find s when u = 2 and r = 4

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LEARNING TASK #1

https://forms.gle/6cgeysrXCCyy41CV9

LEARNING TASK #2

https://forms.gle/j9qBThSkFWY3S7zp6

LEARNING TASK #3

ASSIGNMENT

Performance Task 1

• Give 5 real life examples of direct variations
• Give 5 real life examples of inverse variations
• Give 5 real life examples of joint variations
• Give 5 real life examples of combined variations

THANK YOU!!

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