Indicies - Explanation, Examples and Laws of Indicies

                                          Variation.

In mathematics, variation explains how one quantity changes in relation to another.
It helps us understand relationships like how time affects distance, or how speed affects fuel usage.

There are four main types of variation: direct, inverse, joint, and partial (combined) variation. Let’s go through each type with detailed examples.

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1. Direct Variation

In direct variation, one quantity increases (or decreases) as the other increases (or decreases).
If $y$ varies directly as $x$, the relationship is:

$$y=kx$$

where $k$ is the constant of variation.

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Example 1

If $y$ varies directly as $x$, and $y = 10$ when $x = 5$, find $y$ when $x = 8$.

Solution:

$$y = kx$$

Substitute $y = 10$, $x = 5$:

$$10 = 5k \Rightarrow k = 2$$

When $x=\mathrm{8:}$

$$y = 2(8) = 16$$

Answer: $y=16$

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Example 2

If $y$ varies directly as $x$ and $y = 18$ when $x = 6$, find $y$ when $x = 10$.

Solution:

$$y = kx \quad \text{so} \quad 18 = 6k \Rightarrow k = 3$$

When $x = 10$:

$$y=3(10)=30$$

Answer: $y=30$

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Example 3

If $y$ varies directly as $x$ and $y=40$ when $x=8$, find $y$ when $x=12.$

Solution:

$$y = kx \Rightarrow 40 = 8k \Rightarrow k = 5$$

When $x=12$:

$$y = 5(12) = 60$$

Answer: $y=60$

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2. Inverse Variation

In inverse variation, one quantity increases while the other decreases such that their product is constant.

$$y = \frac{k}{x}$$

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Example 4

If $y$ varies inversely as $x$, and $y = 12$ when $x=\mathrm{3,\; find }$$y$ when $x = 8$.

Solution:

$$y = \frac{k}{x} \Rightarrow 12 = \frac{k}{3} \Rightarrow k = 36$$

When $x=8$:

$$y = \frac{36}{8} = 4.5$$

Answer: $y=4.5$

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Example 5

If $y$ varies inversely as $x$, and $y=20$ when $x=\mathrm{2,}$ find $y$ when $x=5$.

Solution:

$$y = \frac{k}{x} \Rightarrow 20 = \frac{k}{2} \Rightarrow k = 40$$

When $x=5$:

$$y = \frac{40}{5} = 8$$

Answer: $y=8$

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Example 6

If $y$ varies inversely as $x$, and $y=\mathrm{15\; when }$$x = 4$, find $y$ when $x = 10$.

Solution:

$$15 = \frac{k}{4} \Rightarrow k = 60$$

When $x=10$:

$$y = \frac{60}{10} = 6$$

Answer: $y=6$

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3. Joint Variation

In joint variation, a quantity varies directly as the product of two or more other variables.

$$y = kxz$$

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Example 7

If $y$ varies jointly as $x$ and $z$, and $y = 48$ when $x = 4$ and $z = 3$, find $y$ when $x = 6$ and $z = 5$.

Solution:

$$48 = k(4)(3) \Rightarrow k = 4$$

When $x = 6, z = 5$:

$$y = 4(6)(5) = 120$$

Answer: $y = 120$

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Example 8

If $y$ varies jointly as $x$ and $z$, and $y = 72$ when $x = 3$ and $z = 4$, find $y$ when $x = 5$ and $z = 6$.

Solution:

$$72 = k(3)(4) \Rightarrow k = 6$$

When $x = 5, z = 6$:

$$y = 6(5)(6) = 180$$

Answer: $y = 180$

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4. Partial (Combined) Variation

In partial variation, one quantity varies partly directly and partly inversely with another.

$$y = kx + \frac{c}{x}$$

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Example 9

If $y = 10$ when $x = 2$ and $y = 13$ when $x = 3$, find the relationship between $y$ and $x$.

Solution:

$$y = kx + \frac{c}{x}$$

Substitute $x = 2, y = 10$:

$$10 = 2k + \frac{c}{2} \quad (1)$$

Substitute $x = 3, y = 13$:

$$13 = 3k + \frac{c}{3} \quad (2)$$

Solving these equations gives:

$$k = 4, \quad c = 4$$

Hence,

$$y = 4x + \frac{4}{x}$$

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Example 10

If $y = 14$ when $x = 2$ and $y = 20$when $x = 4$, find $y$ in terms of $x$.

Solution:

$$y = kx + \frac{c}{x}$$

Substitute values:
When $x = 2, y = 14$:

$$14 = 2k + \frac{c}{2} \Rightarrow 28 = 4k + c \quad (1)$$

When $x = 4, y = 20$:

$$20 = 4k + \frac{c}{4} \Rightarrow 80 = 16k + c \quad (2)$$

Subtract (1) from (2):

$$80 - 28 = (16k - 4k) \Rightarrow 52 = 12k \Rightarrow k = \frac{13}{3}$$

Substitute $k = \frac{13}{3}$ into (1):

$$28 = 4\left(\frac{13}{3}\right) + c \Rightarrow 28 = \frac{52}{3} + c \Rightarrow c = \frac{32}{3}$$
$$y = \frac{13}{3}x + \frac{32}{3x}$$

Final Relation:

$$y = \frac{13x + 32/x}{3}$$

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🧮 Assignment

1. $y$ varies directly as $x$. If $y = 12$when $x = 4$, find $y$ when $x = 7.$

2. $y$ varies inversely as $x$. If $y = 9$ when $x = 3$, find $y$ when $x = 6$.

3. $y$ varies jointly as $x$ and $z$. If $y = 45$ when $x = 5$ and $z = 3$, find $y$ when $x = 6$ and $z = 2$.

4. $y$ varies partly as $x$ and partly as $\frac{1}{x}$. If $y = 8$ when $x = 2$, and $y = 11$ when $x = 3$, find $y$ when $x = 4$.

5. The volume $V$ of a gas varies directly as its temperature $T$ and inversely as its pressure $P$. If $V = 400$ when $T = 200$ and $P = 50$, find $V$ when $T = 240$ and $P = 60$.