We cannot solve the first equation until we know at least one of the variables, so let's solve the second equation first to solve for . We therefore get:

With our , we can now find x using the first equation:

We therefore get the correct answer of  and .

Let  be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation , 

.

We can find  by setting :

Therefore .

Set  and solve for :

 kilograms

The general formula for direct proportionality is

where  is the proportionality constant. To find the value of this , we plug in  and

Solve for  by dividing both sides by 12

So .

The general formula for direct proportionality is 

where  is how much money you earned,  is the proportionality constant, and  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 . It is given that you earned $32 by working 4 hours. Plug these values into the formula

Solve for  by dividing both sides by 4.

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

Plug in $48.

Divide both sides by 8

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

We must set up a proportion. Since x varies directly with y, when y is multiplied by 2, x is also multiplied by 2. 26 times 2 is 52.

Direct variation:

Let  be the number of pizzas and  be the number of people attending. Because  is directly proportional to , we have

and from the problem

.

Therefore, solving for  we get:

Now solving for ,

Thus, for 18 people 9 pizzas are needed.

Direct proportionality results when two quantities increase or decrease at the same time. In the formula for resistance, , as  increases so too does . The same is true for .  It helps to think of these variables in terms of numbers. When the numerator of this formula is increased and the denominator remains the same, the overall quotient would be larger. The opposite is true if the numerator is decreased and the denominator remains the same (quotient would be smaller).

Direct proportionality means that as one increases so does the other. The equation that is indicative of this statement is the following:

Direct proportionality means you should use the equation 

The question says that the amount of money earned by the business is directly proportional to the number of driveways shoveled so that means y must be the amount of money earned ($3000) and x must be the amount of driveways shoveled (600). Plugging these values into the equation you should get

Solve for k by doing the opposite of what is being done to k on both sides of the equation. Since k is being multiplied by 600 divide both sides by 600

Each driveway earned the snow shoveling business $5

Breaking down the question one small piece at a time:

"The force from gravity () is directly proportional" means that the gravity term is going to be on one side of the proportional symbol, and everything else is going to be on the other side.

"to the inverse...of the distance" means that the distance will be in the denominator.  As distance gets larger, force will get smaller.

"of the square of the distance" means that the distance will be squared.

Putting it all together gives: