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. 

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. 

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 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 

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 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 subtract  from 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.