To solve this system of equations, multiply each equation by an integer that would allow us to cancel one variable. We then substitute that value back into either original equation to find the other variable. In this case we will cancel the x variable by multiplying each equation by 3 and 4 respectively.

1) 

2) 

Multiply equation 1 by 3 and equation 2 by 4:

Add the new equations:

The result is 

Solve for y:

Now substitute this value back into equation 1:

Solve the second equation for , allowing us to solve using the substitution method.

Substitute for   in the first equation, and solve for .

Now, substitute for  in either equation; we will choose the second. This allows us to solve for .

Now we can write the solution in the notation , or .

Multiply the bottom equation by , then add to the top equation:

Divide both sides by 

When we add the two equations, the  variables cancel leaving us with:

Solving for  we get:

We can then substitute our value for  into one of the original equations and solve for :

We are given

We can solve this by using the substitution method.  Notice that you can plug  from the first equation into the second equation and then get

Add  to both sides

Add 9 to both sides

Divide both sides by 5

So . We can use this value to find y by using either equation. In this case, I'll use .

So the solution is 

Let  be the number of adult tickets sold. Then the number of child tickets sold is .

The amount of money raised from adult tickets is ; the amount of money raised from child tickets is . The sum of these money amounts is , so the amount of money raised can be defined by the following equation:

To find the number of adult tickets sold, solve for :

 adult tickets were sold.

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 

Use the elimination process to determine the value of .

Subtract the second equation with the first equation.  This will eliminate the x-variable in order to find the y-variable.

The equation becomes:  

Use this value and substitute into either the first or second equation and solve for the x-variable.

Let's select the first equation and solve for .

Add six on both sides.

The result will be the same if we substitute  into the second equation.

The answer is:  

In order to solve for , multiply the first equation by four in order to eliminate the fractions in front of the coefficients.

The first equation becomes:

The second equation remains the same:  

Subtract the second equation with the first equation:

The result is:  

Divide by two on both sides.

Simplify both sides.

The answer is:  

In order to solve for the  variable, we will need to use the elimination method.

Subtract the first equation with the second equation in order to eliminate the  variable to solve for .

After subtracting equation one with equation two:

Use this value and either equation to solve for . Let's choose the first equation.

Subtract twelve from both sides.

Simplify both sides.

The answer is: