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. 

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. 

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 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. 

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 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. 

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.

This system can be solved a variety of ways, including graphing. To solve algebraically, write an equation for each of the different trains. We will use y to represent the distance from San Francisco, and x to represent the time since 8AM.

The blue train travels 80 miles per hour, so it adds 80 to the distance from San Francisco every hour. Algebraically, this can be written as .

The green train starts 380 miles away from San Francisco and subtracts distance every hour. This equation should be .

To figure out where these trains' paths will intersect, we can set both right sides equal to each other, since the left side of each is .

add  to both sides

divide both sides by 140

Since we wrote the equation meaning time for , this means that the trains will cross paths after 2.714 hours have gone by. To figure out what time it will be then, figure out how many minutes are in 0.714 hours by multiplying . So the trains intersect after 2 hours and about 43 minutes, so at 10:43AM.

To figure out how far from San Francisco they are, figure out how many miles the blue train could have gone in 2.714 hours. In other words, plug 2.714 back into the equation , giving you an answer of .