Direct Variation Equation

Introduction

Direct variation occurs when one quantity changes in proportion to changes in another quantity. This means that if one quantity increases, the other increases proportionally, and if one decreases, the other decreases proportionally. This relationship is linear, resulting in a straight-line graph.

Moreover, when two quantities are in direct variation, one will be a constant multiple of the other. In this article, we'll look into direct variation, covering its definition, formula, graphing direct variation, and work with a few direct variation examples.

Definition of Direct Variation

When two quantities increase or decrease by the same proportionate factor, they are said to exhibit direct variation. Consequently, an increase in one quantity results in a corresponding increase in the other, while a decrease in one quantity leads to a decrease in the other.

In other words, if the ratio of the first quantity to the second remains constant, the quantities are considered directly proportional, with this constant ratio referred to as the coefficient or constant of proportionality.

Real-Life Examples of Direct Variation

Below are two examples illustrating the concept of direct variation:

Example `1`: Let's consider the relationship between the time taken to travel a certain distance and the speed of travel. When driving at a constant speed, the time it takes to travel a certain distance directly varies with the speed. For instance, if it takes `2` hours to travel `100` miles at a constant speed, it would take `4` hours to travel `200` miles. In this scenario, time and distance are in direct variation with the constant of proportionality being speed.

Example `2`: Imagine a scenario where the number of pages printed is directly proportional to the number of cartridges used. If it takes `1` cartridge of ink to print `500` pages, we would need `3` cartridges to print `1500` similar pages.

Direct Variation Symbol

The direct variation formula establishes a relationship between two quantities, indicating that one is a constant multiple of the other. When two quantities exhibit direct proportionality, they are denoted using the symbol "`\propto`". For instance, if we have two quantities, `x` and `y`, in direct variation, their relationship can be expressed as:

\(y \propto x\)

Direct Variation Formula

The connection between two variables, `x` and `y`, exhibiting direct variation, can be expressed by an equation as follows:

\(y = kx\)

Here, the constant of proportionality, ` k `, is defined as ` k = \frac{y}{x} ` when ` x ` is not equal to zero. Hence, the ratio of these two variables remains constant. 

Alternatively, the direct variation equation can be expressed as ` x = \frac{y}{k} `. This signifies that ` x ` is directly proportional to ` y `, with ` k ` being the constant of proportionality, equivalent to ` \frac{1}{k} `.

Additionally, the ratio of the dependent to the independent variable remains constant in a function representing direct variation. The formula for direct variation between two linearly dependent quantities is as follows:

Identify Direct Variation Using Table

Example `1`: Are the values given in the table below in direct variation?

Solution:

We know that the equation for direct variation is `y = kx`.

Proportionality constant, `k = \frac{y}{x}`.

Now, let’s find the value of `k` for all the values in the table using the formula `k = \frac{y}{x}`.

`k = \frac{7}{1} = 7`

`k = \frac{14}{2} = 7`

`k = \frac{21}{3} = 7`

`k = \frac{28}{4} = 7`

`k = \frac{35}{5} = 7`

Therefore, the proportionality constant across all the pairs of `x` and `y` in the table is the same which is `7`.

Hence, the values in the given table are in direct variation.

Example `2`: Does the table below illustrate direct variation between `x` and `y` values?

Solution:

We know that the equation for direct variation is `y = kx`.

Proportionality constant, `k = \frac{y}{x}`.

Now, let’s find the value of `k` for all the values in the table using the formula `k = \frac{y}{x}`.

`k = \frac{30}{1} = 30`

`k = \frac{100}{2} = 50`

`k = \frac{150}{5} = 30`

`k = \frac{180}{6} = 30`

`k = \frac{42}{7} = 6`

Therefore, the proportionality constant across all the pairs of `x` and `y` in the table is not the same.

Hence, the values in the given table are not in direct variation.

Graph of Direct Variation

In direct variation, the relationship between two quantities forms a straight line on a graph, indicating a linear equation with two variables. This equation is represented as ` y = kx `, where ` k ` is the constant of variation. The ratio of change, ` \frac{\Delta y}{\Delta x} `, is also equivalent to ` k `, signifying the slope of the line. The direct variation graph can be expressed as:

Please note that the graph of a direct variation always passes through the origin `(0,0)`.

Difference Between Direct and Inverse Variation

Direct variation and inverse variation go hand in hand. They describe how two mathematical quantities relate to each other. While direct variation occurs when one quantity increases proportionally as the other increases, inverse variation happens when one quantity increases while the other decreases.

Direct and inverse variation are two forms of proportionality. The distinction between them is outlined in the following table:

Solved Examples

Example `1`: If ` x = 5 ` and ` y = 20 ` follow a direct variation, find the constant of proportionality.

Solution:

As `x` and `y` are in a direct variation thus `y = kx` or `k = \frac{y}{x}`.

`k = \frac{20}{5}`

`k = 4`

Therefore, the constant of proportionality, `k` is `4`.

Example 2: Let `x` and `y` be in direct variation, `x = 7` and `y = 22`. What is the direct variation equation that relates `x` and `y`?

Solution:

As `x` and `y` are in a direct variation thus `y = kx` or `k = \frac{y}{x}`.

`k = \frac{22}{7}`

Substitute the value of `k` in `y = kx`.

`y = \frac{22}{7} x`

Therefore, the direct variation equation is `y = \frac{22}{7} x`.

Example `3`: Beth works for 8 hours and earns `\$160`. Express the direct variation equation representing the scenario.

Solution:

We know the direct variation formula is `y = kx` or `k = \frac{y}{x}`.

Replace `y` with `160` and `x` with `8`.

`k = \frac{160}{8}`

`k = 20`

Substitute the value of `k` in `y = kx`.

`y = 20x`

Therefore, the direct variation equation is `y = 20x`. 

Hence, Beth earns `$20` per hour.

Example `4`: A printer can print `30` pages per minute. How many pages will it print in `10` minutes?

Solution:

The number of pages printed is directly proportional to the number of minutes.

Let the printer print `x` number of pages in `10` minutes.

We can show that the number of minutes is directly proportional to the number of pages by writing:

`\frac{1}{30} = \frac{10}{x}`

By cross multiplication, we get

`x \times 1 = 30 \times 10`

`x = 300`

Therefore, the printer will print `300` pages in `10` minutes.

Example `5`: If `8` identical shirts cost `$64`, what is the cost of `12` such shirts?

Solution:

The number of shirts is directly proportional to the cost.

Let `y` be the cost of `12` shirts. 

We can show that the cost is directly proportional to the number of shirts by writing:

`\frac{64}{8} = \frac{y}{12}`

By cross multiplication, we get

`8 \times y = 64 \times 12`

`y = 8 \times 12`

`y = 96`

Therefore,  the cost of `12` shirts is `\$96`.

Practice Problems

Q`1`. If ` x = 8 ` and ` y = 24 ` follow a direct variation, determine the constant of proportionality.

1.  ` k = 3 `
2.  ` k = 2 `
3.  ` k = 4 `
4.  ` k = 6 `

Answer: a

Q`2`. Emma took `30` minutes to pack `6` gift items.  If Emma continues to pack at the same rate, write the equation for direct variation representing this scenario.

1.  `y = 2x`
2.  `y = 3x`
3. `y = 4x`
4.  `y = 5x`

Answer: d

Q`3`. Let `x` and `y` be in direct variation, `x = 9` and `y = 31`. What is the direct variation equation?

1.  ` y = \frac{31}{9} `
2.  ` y = \frac{9}{31} x `
3.  ` y = \frac{31}{9} x `
4.  ` y = 9x `

Answer: c

Q`4`. If a car travels `50` miles per hour, how far will it travel in `3` hours?  Consider that the car continues to travel at the same speed.

1.  `100` miles  
2.  `150` miles  
3.  `200` miles  
4.  `250` miles

Answer: b

Q`5`. If `20` pounds of apples cost `$30`, what is the cost of `30` pounds of apples?

1.  `$40`
2.  `$45`
3.  `$50`
4.  `$60`

Answer: b

Frequently Asked Questions

Q`1`. What is direct variation?

Answer: Direct variation is a relationship between two variables where one increases (or decreases) as the other increases (or decreases) at a constant rate.

Q`2`. How do you recognize direct variation?

Answer: Direct variation is indicated by a straight line passing through the origin on a graph of the two variables.

Q`3`. What is the equation for direct variation?

Answer: The equation is in the form `y = kx`, where `y` and `x` are the dependent and the independent variables respectively, and `k` is the constant of variation.

Q`4`. How do you find the constant of variation?

Answer: Divide any `y` value by its corresponding `x` value; the result is the constant of variation, `k`.

Q`5`. Can direct variation apply to real-life situations?

Answer: Yes, for example, if the cost of a product increases proportionally with the quantity purchased, it exhibits direct variation.