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  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 

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect. 

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

The slope for the first set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

The slope for the second set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

Notice that both of these lines have the same slope, but different  , which means they will never intersect. 

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect. 

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

The slope for the first set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

The slope for the second set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can subtract  from both sides to solve for :

Our equation for this line is 

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's add  to both sides:

Next, we can subtract  from both sides:

Finally, we can divide  by both sides to solve for 

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both  and  values. 

Now that we have a value for , we can plug that value into one of our equations to solve for 

Our point of intersection for these two lines is  This proves that the two lines made from the two sets of coordinate points do intersect. 

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect. 

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

The slope for the first set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

The slope for the second set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can subtract  from both sides to solve for :

Our equation for this line is 

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's subtract  from both sides:

Next, we can add  to both sides:

Finally, we can divide  by both sides to solve for 

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both  and  values. 

Now that we have a value for , we can plug that value into one of our equations to solve for 

Our point of intersection for these two lines is  This proves that the two lines made from the two sets of coordinate points do intersect. 

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect. 

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

The slope for the first set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can subtract  from both sides to solve for :

Our equation for this line is 

The slope for the second set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's subtract  from both sides:

Next, we can subtract  from both sides:

Finally, we can divide  by both sides to solve for 

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both  and  values. 

Now that we have a value for , we can plug that value into one of our equations to solve for 

Our point of intersection for these two lines is  This proves that the two lines made from the two sets of coordinate points do intersect. 

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect. 

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

The slope for the first set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can subtract  to both sides to solve for :

Our equation for this line is 

The slope for the second set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's subtract  from both sides:

Next, we can subtract  from both sides:

Finally, we can divide  by both sides to solve for 

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both  and  values. 

Now that we have a value for , we can plug that value into one of our equations to solve for 

Our point of intersection for these two lines is  This proves that the two lines made from the two sets of coordinate points do intersect. 

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect. 

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

The slope for the first set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can add  to both sides to solve for :

Our equation for this line is 

The slope for the second set of coordinate points:

Now that we have our slope, the formula is:

To solve for , or the , we can plug in one of the coordinate points for the  and  value:

We can subtract  from both sides to solve for :

Our equation for this line is 

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's add  to both sides:

Next, we can add  to both sides:

Finally, we can divide  by both sides to solve for 

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both  and  values. 

Now that we have a value for , we can plug that value into one of our equations to solve for 

Our point of intersection for these two lines is  This proves that the two lines made from the two sets of coordinate points do intersect.