The general formula for direct proportionality is

$\displaystyle a=kb$

where $\displaystyle k$ is the proportionality constant. To find the value of this $\displaystyle k$, we plug in $\displaystyle a=24$ and $\displaystyle b=12$

$\displaystyle 24=12k$

Solve for $\displaystyle k$ by dividing both sides by 12

$\displaystyle \frac{24}{12}=\frac{12k}{12}$

$\displaystyle 2=k$

So $\displaystyle k=2$.

The general formula for direct proportionality is 

$\displaystyle M=kh$

where $\displaystyle M$ is how much money you earned, $\displaystyle k$ is the proportionality constant, and $\displaystyle h$ is the number of hours worked.

Before we can figure out how many hours you need to work to earn $48, we need to find the value of $\displaystyle k$. It is given that you earned $32 by working 4 hours. Plug these values into the formula

$\displaystyle 32=4k$

Solve for $\displaystyle k$ by dividing both sides by 4.

$\displaystyle \frac{32}{4}=\frac{4k}{4}$

$\displaystyle 8=k$

So $\displaystyle k=8$. We can use this to find out the hours you need to work to earn $48. With $\displaystyle k=8$, we have

$\displaystyle M=8h$

Plug in $48.

$\displaystyle 48=8h$

Divide both sides by 8

$\displaystyle \frac{48}{8}=\frac{8h}{8}$

$\displaystyle 6=h$

So you will need to work 6 hours to earn $48.

If $\displaystyle V,P,T$ are the volume, pressure, and temperature, then the variation equation will be, for some constant of variation $\displaystyle K$,

$\displaystyle V = \frac{KT}{P}$

To calculate $\displaystyle K$, substitute $\displaystyle V = 100, T = 310, P = 1,020$:

$\displaystyle 100 = \frac{K \cdot 310}{1,020}$

$\displaystyle \frac{K \cdot 310}{1,020} \cdot \frac{1,020}{310} = 100 \cdot \frac{1,020}{310}$

$\displaystyle K = 329.03$

The variation equation is 

$\displaystyle V = \frac{329.03 T}{P}$

so substitute $\displaystyle T =290, P = 900$ and solve for $\displaystyle V$. 

$\displaystyle V = \frac{329.03 \cdot 290}{900} \approx 106 \textrm{ m}^{3}$

The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as $\displaystyle M=kC$. M is the monthly cost, C is the number of cars owned, and k is the constant of variation.

Given that it costs $420 a month to insure 3 cars, we can find the k-value.

$\displaystyle 420=k * 3$

Divide both sides by 3.

$\displaystyle k=140$

Now, we have the mathematical relationship.

$\displaystyle M=140C$

Finding how much it costs to insure 5 cars can be found by substituting 5 for C and solving for M.

$\displaystyle M=140(5)$

$\displaystyle M=700$

Direct Variation is a relationship that can be represented by a function in the form

$\displaystyle y = kx$, where $\displaystyle k \neq 0$

$\displaystyle k$ is the constant of variation for a direct variation. $\displaystyle k$ is the coefficient of $\displaystyle x$.

$\displaystyle 7y = 2x$

$\displaystyle y = \frac{2}{7}x$

The equation is in the form $\displaystyle y = kx$, so the equation is a direct variation.

The constant of variation or $\displaystyle k$ is $\displaystyle \frac{2}{7}$

Therefore, the answer is,

Yes it is a direct variation, $\displaystyle y = \frac{2}{7}x,$ with a direct variation of $\displaystyle \frac{2}{7}$

The general formula for direct proportionality is

$\displaystyle x=ky$

where $\displaystyle k$ is our constant of proportionality. From here we can plug in the relevant values for $\displaystyle x$ and $\displaystyle y$ to get

$\displaystyle 24=4k$

Solving for $\displaystyle k$ requires that we divide both sides of the equation by $\displaystyle 4$, yielding

$\displaystyle k=6$

Because the cost varies directly with the number of people attending, we have the equation

$\displaystyle C=kN$

Where $\displaystyle C$ is the cost and $\displaystyle N$ is the number of people attending. 

We solve for $\displaystyle k$, the constant of variation, by plugging in $\displaystyle C=100$ and $\displaystyle N=20$.

$\displaystyle 100=k(20)$

And by dividing by 20 on both sides

$\displaystyle \frac{100}{20}=\frac{k(20) }{20}$

Yields

$\displaystyle k=5$

Write the formula for direct proportionality.

$\displaystyle y=kx$

Let:

$\displaystyle x=\textup{hours worked}$

$\displaystyle y= \textup{earnings}$

Substitute twelve dollars and eight hours into this equation to solve for $\displaystyle k$.

$\displaystyle \$12=k(8)$

Divide by eight on both sides.

$\displaystyle k=\frac{12}{8} = \frac{3}{2}$

Substitute $\displaystyle k$ back into the formula.

$\displaystyle y=\frac{3}{2}x$

To find out the minimum number of hours Billy must work to make $\displaystyle \$100$, substitute $\displaystyle \$100$ into $\displaystyle y$ and solve for $\displaystyle x$.

$\displaystyle \$100=\frac{3}{2}x$

Multiply by two thirds on both sides.

$\displaystyle 100\cdot \frac{2}{3}=\frac{3}{2}x \cdot \frac{2}{3}$

Simplify both sides.

$\displaystyle x=\frac{200}{3} = 66.\overline{6}$

Billy must work at least $\displaystyle 67$ hours to earn as much required.

$\displaystyle f\left ( x \right ) = x^{2}-3$

which is the same as

$\displaystyle y = x^{2}-3$

If we solve for $\displaystyle x$ we get

$\displaystyle x^{2} = y +3$

Taking the square root of both sides gives us the following:

$\displaystyle x = \pm \sqrt{y+3}$

Interchanging $\displaystyle x$ and $\displaystyle y$ gives us

$\displaystyle y = \pm \sqrt{x + 3}$

Which is not one-to-one and therefore not a function.

Starting with $\displaystyle f\left ( x \right ) = 2x+3$

Replace $\displaystyle x$ with $\displaystyle g\left ( x \right )$.

We get the following:

$\displaystyle 2\left ( \frac{x-3}{2} \right )+3=x-3+3$

Which is equal to $\displaystyle x$.