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Chapter 0: Problem 1

Write an equation that describes each variation. Use k as the constant of variation. \(y\) varies directly with \(x\).

Short Answer

Expert verified
The equation is \(y = kx\).

Step by step solution

01

Understanding Direct Variation

When we say that one quantity varies directly with another, it means that they are proportional to each other. Specifically, if we have a variable \(y\) that varies directly with \(x\), then \(y\) is equal to a constant \(k\) multiplied by \(x\). This relationship is written as \(y = kx\).
02

Writing the Direct Variation Equation

Since \(y\) varies directly with \(x\), we use the equation for direct variation: \(y = kx\). Here, \(k\) is a constant that represents the ratio of \(y\) to \(x\) and does not change as \(x\) and \(y\) change.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportionality
Proportionality is a key concept in mathematics, often used to describe how two quantities change in relation to each other. When two quantities are proportional, their ratio remains constant. This means if one quantity doubles, the other also doubles, maintaining the same ratio or relationship. In the context of the exercise, since we have a direct variation, the relationship between the variables is fully proportional. In simple terms, if you increase one value, the other increases at the same rate.
  • This proportional relationship is expressed in the form of an equation: directly as \(y = kx\), where \(k\) is the constant of proportionality.
  • This constant \(k\) indicates how much \(y\) increases when \(x\) increases by one unit.
  • The concept of proportionality is crucial to understanding phenomena where changes in one variable predict changes in another.
Constant of Variation
The Constant of Variation, often represented by \(k\), is a fundamental component of the direct variation equation. It represents the strength and direction of the relationship between two proportional variables. Think of it as the fixed number that scales the variable \(x\) to get the value of the variable \(y\).
  • In the equation \(y = kx\), \(k\) tells us how many units \(y\) changes for each unit change in \(x\).
  • If \(k = 2\), it means that for every 1 unit increase in \(x\), \(y\) increases by 2 units.
  • The constant of variation is essential for determining the specific change one variable has on another in proportional relationships.

Knowing the value of \(k\) allows us to predict the effect of changes in \(x\) on \(y\) and this can be a very powerful tool in analyzing and predicting behavior in direct variation problems.
Linear Relationship
A linear relationship is one of the simplest forms of relationship between two variables, represented by a straight line when graphed. In a direct variation scenario, the relationship between \(x\) and \(y\) is linear, meaning that if you plot the values of \(x\) against \(y\), they will form a straight line through the origin.
  • The line's slope is determined by the constant of variation \(k\).
  • A positive \(k\) results in a line that slopes upwards (increases) as \(x\) increases, while a negative \(k\) would indicate the line slopes downwards (decreases).
  • The straightforward nature of linear relationships makes them easy to work with, as changes in one variable lead to predictable and consistent changes in the other.

Understanding linear relationships helps in solving problems involving direct variation and makes it easier to visualize how two variables connect as you calculate or graph based on their equations.

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Most popular questions from this chapter

Rick and Mike are roommates and leave Gainesville on Interstate 75 at the same time to visit their girlfriends for a long weekend. Rick travels north and Mike travels south. If Mike's average speed is 8 mph faster than Rick's, find the speed of each if they are 210 miles apart in 1 hour and 30 minutes.

Write a quadratic equation in standard form that has the solution set \(\\{2,5\\} .\) Alternate solutions are possible.

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Refer to the following: Einstein's special theory of relativity states that time is relative: Time speeds up or slows down, depending on how fast one object is moving with respect to another. For example, a space probe traveling at a velocity \(v\) near the speed of light \(c\) will have "clocked" a time \(t\) hours, but for a stationary observer on Earth that corresponds to a time \(t_{0} .\) The formula governing this relativity is given by $$ t=t_{0} \sqrt{1-\frac{v^{2}}{c^{2}}} $$ If the time elapsed on a space probe mission is 18 years but the time elapsed on Earth during that mission is 30 years, how fast is the space probe traveling? Give your answer relative to the speed of light.
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