DIRECT VARIATION. MATHEMATICS 9. undefined. 

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[Audio] Direct variation is an important concept in mathematics, especially when it comes to problem solving. In direct variation, two variables increase or decrease together in a proportional relationship. To solve such a problem, it is necessary to figure out the constant of proportionality, which is the multiplier between the variables. This can be done by looking at a given example and solving for the unknown. For instance, if given that y varies directly with x, and y equals 8 when x equals 4, then the constant of proportionality is 2, since 8 is twice 4. This indicates that when x doubles, y will also double.. 

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PROBLEM SOLVING INVOLVING
DIRECT VARIATION

[Audio] We will be discussing direct variation, which happens when two quantities are directly proportional to each other. An example of this is when the first quantity doubles, so does the second quantity. Similarly, if one quantity is halved, so is the other. We will delve deeper into the concept of direct variation today.. 

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Direct variation is a type of proportionality wherein one quantity directly varies with respect to a change in another quantity.

[Audio] Slide 4 deals with the topic of direct variation. This is a mathematical relationship between two quantities where one has a constant ratio to the other. This is typically represented as an equation, where one of the terms is the dependent variable whilst the other is the independent variable. This is often shown as y = kx, where ‘y’ is dependent on ‘x’ and ‘k’ is the constant of variation. As x increases, y increases in the same ratio and when x decreases, y decreases in the same ratio too. This concept can also be used for other types of relationships, such as when two quantities vary inversely.. 

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Study the following statements: 
“? ?????? ???????? ?? ?” 
“? ?? ???????? ???????????? ?? ?” 
“? ?? ???????????? ?? ?” 
(? is the dependent variable, ? is the independent variable) may be translated mathematically as ? = ??, where ? is the constant of variation. 
For two quantities, ? ??? ?, an increase in ? causes an increase in ? as well.
Similarly, a decrease in ? causes a decrease in ?.

[Audio] We can observe how two quantities can vary directly with each other from this example. The cost of fish per kilogram (c) is directly proportional to its weight in kilograms (w). The equation c=kw can be used, where k is the constant of proportionality. We can find the value of k by rearranging the equation to k=c/w. It is important to note that the cost (c) and the weight (w) are directly proportional, meaning that if the weight increases, the cost increases and vice versa.. 

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EXAMPLE:
The cost of fish per kilograms (c) varies directly as its weight in kilograms (w).
c=kw

[Audio] We are going to tackle one of the more useful types of equations known as Direct Variations. These equations indicate that when one of the variables increases, the other one increases/decreases accordingly. To solve them, we must first express the statement into an equation or formula for direct variation. Then we substitute the given values for the variables, solve for the constant of variation, and write the equation which satisfies x and y. Finally, we substitute the remaining values and find the unknown.. 

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Steps in solving problem involving Direct Variations

Express the statement into equation or formula for direct variation
Substitute the given values for the variables.
Solve for the constant of variation.
Write the equation which satisfies x and y.
Substitute the remaining values and find the unknown.

[Audio] We will be examining direct variation. This is a connection in which two variables directly affect each other. If one variable goes up, the other will increase too; if one decreases, the other will do the same. To find direct variations, we use an equation with two existing values to discover the unknown constant of proportionality. Once the constant of proportionality is identified, it can be used to discover the value of one of the variables when the other is given. Let's look at an example.. 

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Direct variation word problems may be solved using the following steps:

1) Recognizing the word problem consists of direct variation as exemplified by presence of verbiage listed above;

2) Writing an equation containing known values for both variables resulting in an expression containing only the unknown representing the constant of proportionality;

3) Solving the equation for the constant of proportionality;

4) Using the calculated constant of proportionality to determine the value of one of the variables given the other.

[Audio] Today, we are exploring the concept of direct variation, which says that when the value of one quantity increases, so does the value of another. For this example, we're looking at the relationship between the amount of money raised at a charity fundraiser and the number of attendees. We can see from the given situation that when there were five attendees, the amount of money raised was $100. We'll be using this example to explore the concept of direct variation in mathematics.. 

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Situation/Problem:
The amount of money raised at a charity fundraiser is directly proportional to the number of attendees.  The amount of money raised for five attendees was $100.

[Audio] Today, we'll be looking at direct variation - variations that have a constant ratio between two variables. For example, if 'x' increases by 4, then 'y' increases by 8. We can say then that 'y' is directly proportional to 'x'. Our question today is to identify the type of variation in a given statement. Let's look at some examples and see if we can answer this question.. 

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QUESTION 1: Identify the type of variation in a given statement

[Audio] Direct Variation is a mathematical relationship between two variables, wherein one is proportional to the other; as one increases in value, the other also increases in value. This can be seen in the statement: 'X is directly proportional to Y', wherein a change in the value of X will cause a corresponding change in the value of Y, and vice versa. It is essential to understand this type of relationship in order to be able to solve various mathematical problems.. 

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[Audio] We need to solve a Direct Variation problem, where the given values of x (number of attendees) and y (dollar amount raised) are related. Knowing the number of attendees is equal to 60, we can calculate the dollar amount raised.. 

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QUESTION 2:  When x = 60 attendees, what is dollar amount raised?

[Audio] Hei! Vi skal se på direkte variasjon fra et matematikk 9 perspektiv. Direkte variasjon betyr at det er et direkte proporsjonalt forhold mellom de to variablene. Vi kan uttrykke det med uttrykket kx. Det betyr at den avhengige variablen y, som er mengden av pengene som skal samles inn, er direkte proporsjonalt med antallet deltakere, som er den uavhengige variablen x. Når vi får en gitt verdi på 5 deltakere og en verdi på 100 dollar som samles inn, kan vi regne ut konstanten k. Vi kan sette det inn i uttrykket kx og finne at k, eller konstanten, er 20. Når vi vet dette kan vi også regne ut hvor mye penger som skal samles inn når det er 60 deltakere. Vi kan bare sette det inn i uttrykket kx for å finne ut at summen som skal samles inn er 1200 dollar.. 

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y = kx
Knowing $100 was raised by five attendees:
100 = 5k
5k = 100
k = 20 

When x = 60 attendees, what is dollar amount raised?
y = kxy = 20x
y = 20(60)
y = $1,200

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DIRECT VARIATION

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Direct variation is an important concept in mathematics, especially when it comes to problem solving. In direct variation, two variables increase or decrease together in a proportional relationship. To solve such a problem, it is necessary to figure out the constant of proportionality, which is the multiplier between the variables. This can be done by looking at a given example and solving for the unknown. For instance, if given that y varies directly with x, and y equals 8 when x equals 4, then the constant of proportionality is 2, since 8 is twice 4. This indicates that when x doubles, y will also double.

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We will be discussing direct variation, which happens when two quantities are directly proportional to each other. An example of this is when the first quantity doubles, so does the second quantity. Similarly, if one quantity is halved, so is the other. We will delve deeper into the concept of direct variation today.

Slide 4 deals with the topic of direct variation. This is a mathematical relationship between two quantities where one has a constant ratio to the other. This is typically represented as an equation, where one of the terms is the dependent variable whilst the other is the independent variable. This is often shown as y = kx, where ‘y’ is dependent on ‘x’ and ‘k’ is the constant of variation. As x increases, y increases in the same ratio and when x decreases, y decreases in the same ratio too. This concept can also be used for other types of relationships, such as when two quantities vary inversely.

We can observe how two quantities can vary directly with each other from this example. The cost of fish per kilogram (c) is directly proportional to its weight in kilograms (w). The equation c=kw can be used, where k is the constant of proportionality. We can find the value of k by rearranging the equation to k=c/w. It is important to note that the cost (c) and the weight (w) are directly proportional, meaning that if the weight increases, the cost increases and vice versa.

We are going to tackle one of the more useful types of equations known as Direct Variations. These equations indicate that when one of the variables increases, the other one increases/decreases accordingly. To solve them, we must first express the statement into an equation or formula for direct variation. Then we substitute the given values for the variables, solve for the constant of variation, and write the equation which satisfies x and y Finally, we substitute the remaining values and find the unknown.

We will be examining direct variation. This is a connection in which two variables directly affect each other. If one variable goes up, the other will increase too; if one decreases, the other will do the same. To find direct variations, we use an equation with two existing values to discover the unknown constant of proportionality. Once the constant of proportionality is identified, it can be used to discover the value of one of the variables when the other is given. Let's look at an example.

Today, we are exploring the concept of direct variation, which says that when the value of one quantity increases, so does the value of another. For this example, we're looking at the relationship between the amount of money raised at a charity fundraiser and the number of attendees.  We can see from the given situation that when there were five attendees, the amount of money raised was $100. We'll be using this example to explore the concept of direct variation in mathematics.

Today, we'll be looking at direct variation - variations that have a constant ratio between two variables. For example, if 'x' increases by 4, then 'y' increases by 8. We can say then that 'y' is directly proportional to 'x'. Our question today is to identify the type of variation in a given statement. Let's look at some examples and see if we can answer this question.

Direct Variation is a mathematical relationship between two variables, wherein one is proportional to the other; as one increases in value, the other also increases in value. This can be seen in the statement: 'X is directly proportional to Y', wherein a change in the value of X will cause a corresponding change in the value of Y, and vice versa. It is essential to understand this type of relationship in order to be able to solve various mathematical problems.

We need to solve a Direct Variation problem, where the given values of x (number of attendees) and y (dollar amount raised) are related. Knowing the number of attendees is equal to 60, we can calculate the dollar amount raised.

Hei! Vi skal se på direkte variasjon fra et matematikk 9 perspektiv. Direkte variasjon betyr at det er et direkte proporsjonalt forhold mellom de to variablene. Vi kan uttrykke det med uttrykket kx. Det betyr at den avhengige variablen y, som er mengden av pengene som skal samles inn, er direkte proporsjonalt med antallet deltakere, som er den uavhengige variablen x Når vi får en gitt verdi på 5 deltakere og en verdi på 100 dollar som samles inn, kan vi regne ut konstanten k Vi kan sette det inn i uttrykket kx og finne at k, eller konstanten, er 20. Når vi vet dette kan vi også regne ut hvor mye penger som skal samles inn når det er 60 deltakere. Vi kan bare sette det inn i uttrykket kx for å finne ut at summen som skal samles inn er 1200 dollar.