cross multiply:

(6)(19) = 9x

114=9x

x = 38/3

Cross multiply:

Let numerator = N and denominator = D.

According to the first statement, 

N = (D x 5) + 4.

According to the second statement, N / 2 = 3 * D. 

Let's multiply the second equation by –2 and add itthe first equation:

–N = –6D

+[N = (D x 5) + 4]

=

–6D + (D x 5) + 4 = 0

–1D + 4 = 0

D = 4

Thus, N = 24.

Therefore, N/D = 24/4 = 6.

Solve for :

Step 1: Cross-multiply to eliminate the fraction

Step 2: Solve for :

Since we know that  stands for weeks, the answer has to have something to do with the weeks. This eliminates "the number of candles she has remaining for the month." Also, we can eliminate "the number of weeks that she has sold candles this month" because that would be our value for , not what we'd multiply  by. The correct answer is,  "the number of candles that she sells each week."

First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as

where  represents Joule's age and  is Newton's age.

The statement, "Newton is Toby's age younger than eleven years" is translated as

where  is Toby's age.

The third statement, "Toby is one year younger than Joule" is

.

So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get

Plug this equation into the first equation to get

Solve for . Add  to both sides

Divide both sides by 3

So Joules is 9 years old. Plug this value into the third equation to find Toby's age

Toby is 8 years old. Use this value to find Newton's age using the second equation

Now, we have the age of the following dogs:

Joule: 9 years

Newton: 3 years

Toby: 8 years

Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly"  and "completing assignments" , then we can easily construct a simple system of equations, 

and 

.

We can multiply the first equation by  to yield .

This allows us to cancel the  terms when we add the two equations together. We get , or .

A quick substitution tells us that . So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.

Let's call "" the cost of one jersey (this is the value we want to find)

Let's call the amount of money Jack starts with ""

Let's call the amount of money Aaron starts with ""

We know Jack buys a glove for  and  bats for  each, and then has  left over after. Thus:

simplifying,  so Jack started with 

We know Aaron buys  hats for  each and  jerseys (unknown cost "") and spends all his money.

The last important piece of information from the problem is Aaron starts with  dollars more than Jack. So:

From before we know:

Plugging in:

so Aaron started with 

Finally we plug  into our original equation for A and solve for x:

Thus one jersey costs 

6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets, .

Therefore, we can say .

The amount of money raised from adult tickets is $11 per ticket mutiplied by  tickets, or  dollars; similarly,  dollars are raised from child tickets. Add these together to get the total amount of money raised:

These two equations form our system of equations.

The first step is to square each side of the inequality.

Now simplify each side.

Now subtract the left side of the inequality to make it zero, so that we can use the quadratic formula.

Now we can use the quadratic formula.

Recall the quadratic formula.

Where , , and , correspond to coefficients in the quadratic equation.

In this case  ,  , and .

Now plug these values into the quadratic equation, and we get.

Now since we are dealing with an inequality, we put the least value on the left side, and the greatest value on the right. It will look like the following.