Direct Variation

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Question 1 of 3

1. Question
Which of the following is a direct variation?
- 1. $y = 3 x$ $y=3x$
- 2. $y = \frac{3}{x}$ $y=3/x$
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Direct Variation

A relationship between two variables, where if one increases the other also increases. Similarly, if one variable decreases the other also decreases.

First, list down a few values for $x$ $x$ and substitute it to the equations to compute for the corresponding $y$ $y$ values

First equation:

$x$ $x$ $1$ $1$ $2$ $2$ $3$ $3$

$y$ $y$

Second equation:

$x$ $x$ $1$ $1$ $2$ $2$ $3$ $3$

$y$ $y$

$y$ $y$ $=$ $=$ $3$ $3$ $x$ $x$

$y$ $y$ $=$ $=$ $\frac{3}{\color{#004ec4}{x}}$

$y$ $y$ $=$ $=$ $3$ $3$ $(1)$ $(1)$

$y$ $y$ $=$ $=$ $3$ $3$

$y$ $y$ $=$ $=$ $\frac{3}{\color{#004ec4}{1}}$

$y$ $y$ $=$ $=$ $3$ $3$

$y$ $y$ $=$ $=$ $3$ $3$ $(2)$ $(2)$

$y$ $y$ $=$ $=$ $6$ $6$

$y$ $y$ $=$ $=$ $\frac{3}{\color{#004ec4}{2}}$

$y$ $y$ $=$ $=$ $\frac{3}{2}$ $3/2$

$y$ $y$ $=$ $=$ $3$ $3$ $(3)$ $(3)$

$y$ $y$ $=$ $=$ $9$ $9$

$y$ $y$ $=$ $=$ $\frac{3}{\color{#004ec4}{3}}$

$y$ $y$ $=$ $=$ $1$ $1$

$x$ $x$ $1$ $1$ $2$ $2$ $3$ $3$

$y$ $y$ $3$ $3$ $6$ $6$ $9$ $9$

$x$ $x$ $1$ $1$ $2$ $2$ $3$ $3$

$y$ $y$ $3$ $3$ $\frac{3}{2}$ $3/2$ $1$ $1$

Notice that on the first equation, as the $x$ $x$ value increases, the $y$ $y$ value also increases.

Hence, the first equation, ( $y = 3 x$ $y=3x$ ) is a direct variation.

It is also worth noting that in the second equation, as the $x$ $x$ value increases, the $y$ $y$ value decreases. This is an inverse variation.

$y = 3 x$ $y=3x$

Question 2 of 3

A car travels

600

$600$ km in

8

$8$ hours. How far would it travel in

11

$11$ hours, given that this is a direct variation?

(825) $k m$ $km$

Question 3 of 3

The time taken,

T

$T$ , for a pendulum to swing varies directly at the square root of its length. If one swing of a

100

$100$ -centimetre pendulum takes

2

$2$ seconds, find the time taken for one swing of a

36

$36$ centimetre pendulum.

Write answer in decimal form

(1.2) seconds

$y$ $y$	$=$ $=$	$k$ $k$ $x$ $x$	Direct Variation Formula
$600$ $600$	$=$ $=$	$k$ $k$ $(8)$ $(8)$	Substitute known values
$600$ $600$	$=$ $=$	$8 k$ $8k$
$600$ $600$ $\div 8$ $divide 8$	$=$ $=$	$8 k$ $8k$ $\div 8$ $divide 8$	Divide both sides by $8$ $8$
$75$ $75$	$=$ $=$	$k$ $k$
$k$ $k$	$=$ $=$	$75$ $75$

$T$ $T$	$=$ $=$	$k$ $k$ $\sqrt{L}$ $sqrtL$	New formula
$2$ $2$	$=$ $=$	$k$ $k$ $\sqrt{100}$ $sqrt100$	Substitute known values
$2$ $2$	$=$ $=$	$10 k$ $10k$
$2$ $2$ $\div 10$ $divide 10$	$=$ $=$	$10 k$ $10k$ $\div 10$ $divide 10$	Divide both sides by $10$ $10$
$0.2$ $0.2$	$=$ $=$	$k$ $k$
$k$ $k$	$=$ $=$	$0.2$ $0.2$

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Quiz summary

Information

1. Question

Hint

Nice Job!

Direct Variation

2. Question

Hint

Excellent!

Direct Variation Formula

Remember

3. Question

Hint

Fantastic!

Direct Variation Formula

Remember

Quizzes

$y$ $y$	$=$ $=$	$\frac{3}{\color{#004ec4}{1}}$

$y$ $y$	$=$ $=$	$3$ $3$

$y$ $y$	$=$ $=$	$\frac{3}{\color{#004ec4}{2}}$

$y$ $y$	$=$ $=$	$\frac{3}{2}$ $3/2$

$y$ $y$	$=$ $=$	$\frac{3}{\color{#004ec4}{3}}$

$y$ $y$	$=$ $=$	$1$ $1$

$x$ $x$	$1$ $1$	$2$ $2$	$3$ $3$
$y$ $y$	$3$ $3$	$\frac{3}{2}$ $3/2$	$1$ $1$

$y$ $y$	$=$ $=$	$k x$ $kx$
$y$ $y$	$=$ $=$	$75 x$ $75x$	Substitute $k$ $k$

$T$ $T$	varies directly	at the square root of its length
$T$ $T$	$=$ $=$	$k \sqrt{L}$ $ksqrtL$	Insert $k$ $k$ , the constant of variation

$T$ $T$	$=$ $=$	$0.2 \sqrt{L}$ $0.2sqrtL$	Updated formula
$T$ $T$	$=$ $=$	$0.2 \sqrt{36}$ $0.2sqrt36$	Substitute $36$ $36$ cm
$T$ $T$	$=$ $=$	$0.2 (6)$ $0.2(6)$
$T$ $T$	$=$ $=$	$1.2$ $1.2$