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Chapter 7: Problem 17

Fill in each blank with the correct response. (a) If the constant of variation is positive and \(y\) varies directly as \(x,\) then as \(x\) increases, \(y=\) ____. (increases/decreases) (b) If the constant of variation is positive and \(y\) varies inversely as \(x,\) then as \(x\) increases, \(V\) ____. (increases/decreases)

Short Answer

Expert verified
a) increases, b) decreases

Step by step solution

01

Understanding Variation

First, recognize the types of variation. Direct variation means that as one variable increases, the other also increases. Inverse variation means that as one variable increases, the other decreases.
02

Analyze Direct Variation

If the constant of variation is positive and y varies directly as x, we can use the equation: \[ y = kx \] where \( k \) is the constant of variation. As \( x \) increases, \( y \) will also increase because multiplying \( x \) by a positive constant \( k \) yields a greater value for \( y \).
03

Fill in the Blank for Direct Variation

Based on the analysis in Step 2, when \( x \) increases for direct variation, \( y \) increases. Fill in the blank: 'a) If the constant of variation is positive and \( y \) varies directly as \( x \), then as \( x \) increases, \( y \) = increases.'
04

Analyze Inverse Variation

If the constant of variation is positive and y varies inversely as x, we can use the equation: \[ y = \frac{k}{x} \] where \( k \) is the constant of variation. As \( x \) increases, \( y \) will decrease because dividing a positive constant \( k \) by a larger \( x \) yields a smaller value for \( y \).
05

Fill in the Blank for Inverse Variation

Based on the analysis in Step 4, when \( x \) increases for inverse variation, \( y \) decreases. Fill in the blank: 'b) If the constant of variation is positive and \( y \) varies inversely as \( x \), then as \( x \) increases, \( y \) = decreases.'

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Key Concepts

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

Direct Variation
When we talk about direct variation, we're talking about a relationship where two variables, let's call them \( x \) and \( y \), increase or decrease together. If \( y \) varies directly as \( x \), it means that as \( x \) gets larger, so does \( y \). This relationship can be represented by the equation \( y = kx \), where \( k \) is a constant called the constant of variation.
  • If \( k \) is positive, as \( x \) increases, \( y \) increases too.
  • If \( k \) is negative, as \( x \) increases, \( y \) decreases.
To break it down simply, in a direct variation:
  • Bigger \( x \) means bigger \( y \) if \( k \) is positive.
  • Smaller \( x \) means smaller \( y \) if \( k \) is positive.
Think of direct variation as a straightforward, predictable pattern where both variables move in the same direction.
Inverse Variation
Inverse variation is a bit different from direct variation. Here, as one variable increases, the other decreases. If \( y \) varies inversely as \( x \), this means that as \( x \) gets larger, \( y \) gets smaller. The relationship can be described by the equation \( y = \frac{k}{x} \), where \( k \) is a positive constant. In simpler terms:
  • When \( x \) goes up, \( y \) comes down.
  • When \( x \) goes down, \( y \) goes up.
This might sound tricky at first, but it's a very common concept. For instance, if you drive faster (increase your speed, \( x \)), the time it takes to reach your destination (\( y \)) decreases. Or think about sharing a cake: more friends sharing the cake (increasing \( x \)) means each friend gets a smaller piece (decreasing \( y \)).
Constant of Variation
The constant of variation is a key part of understanding both direct and inverse variations. This constant, represented by \( k \), determines how \( x \) and \( y \) relate to one another. In the formula for direct variation, \( y = kx \), \( k \) tells us how much \( y \) changes for a given change in \( x \).
For inverse variation, the formula is \( y = \frac{k}{x} \), and \( k \) also tells us how much \( y \) changes as \( x \) changes. If \( k \) is:
  • Positive and big, the relationship is stronger.
  • Positive and small, the relationship is weaker.
  • Negative, the variables will change in opposite directions for direct variations.
Understanding \( k \) helps you predict how changes in one variable will affect the other. It's like the glue that binds \( x \) and \( y \) together in these relationships.
Algebraic Equations
Algebraic equations are the foundation for understanding direct and inverse variations. An equation is a mathematical statement that shows the equality between two expressions. In the context of variation:
  • For direct variation, the algebraic equation is \( y = kx \).
  • For inverse variation, it's \( y = \frac{k}{x} \).
Learning to manipulate these equations can help you solve many types of problems. For example:
  • In a direct variation problem, if you know the values of \( y \) and \( x \), you can find \( k \) by rearranging the formula to \( k = \frac{y}{x} \).
  • In an inverse variation problem, if you know \( y \) and \( x \), rearranging the formula gives you \( k = yx \).
Mastering these equations means you can tackle a variety of algebra problems with confidence!

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Most popular questions from this chapter

Simplify by starting at "the bottom" and working upward. $$ 7-\frac{3}{5+\frac{2}{4-2}} $$

Solve each variation problem. The pressure exerted by water at a given point varies directly with the depth of the point beneath the surface of the water. Water exerts 4.34 lb per in. 2 for every 10 ft traveled below the water's surface. What is the pressure exerted on a scuba diver at \(20 \mathrm{ft} ?\)

Solve each variation problem. For a body falling freely from rest (disregarding air resistance), the distance the body falls varies directly as the square of the time. If an object is dropped from the top of a tower \(400 \mathrm{ft}\) high and hits the ground in \(5 \mathrm{sec}\), how far did it fall in the first \(3 \mathrm{sec} ?\)

Solve each equation for \(k\) $$ 200=15 k $$

Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$
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