The break-even point is where the costs equal the revenue.

Costs: 

Revenue: 

So Costs = Revenue or

Solving for  shows the break-even point is achieved when 15 pairs of boots are sold in a month.

Profits = Revenue - Costs

Revenue: 

Costs: 

So

Solving:

for means that 20 pairs of boots must be sold to make $200 profit.

This problem can be solved by using substitution.  Write the first equation  in terms of  and substitute it into the second equation.

So  and thus  and solving for  and then .

So the sum of  and  is 7.

We have 3 unkown variables and only 2 equations. Instead of trying to solve for   or , notice that we can substitute 2 in for the entire expression :

Substitution gives:

Subtract 8 from both sides: 

Divide by 3:

We need 2 equations, because we have 2 unkown variables. Let  = the entrance fee, and  = the fee per ride. One ride costs  dollars. We know that Lisa spent 120 dollars in total. Since Lisa went on 6 rides, she spent  dollars on rides. Her only other expense was the entrance fee, : 

Apply similar logic to Tom:

Subtracting the second equation from the first equation results in:

Divide both sides by 2:

So every ride costs 12.5 dollars. Plugging 12.5 back into one of the original equations allows us to solve for the entrance fee:

Subtract 50 from both sides:

For this system, it will be easiest to solve by substitution. The  variable is already isolated in the second equation. We can replace  in the first equation with , since these two values are equal.

Now we can solve for .

Now that we know the value of , we can solve for  by using our original second equation.

The final answer will be the ordered pair .

Solve the system of equations using substitution.

First, isolate one of the variables. Since we are solving for , we are going to isolate  in the second equation. 

Replace with in our first equation.

Now we can solve to isolate .

After 20 minutes the first train would have traveled 20 miles. Let x be the amount of time elapsed. When 20 + 60x = 100x you will have the time in hours. 20 = 40x, x = 0.5 hrs. 0.5 hrs = 30 minutes. 

Solve by method of elimination. Multiply the first equation by 8 to eliminate the variable, m. Our first equation will then become .

By adding this new equation

with our second equation

We will see that our m cancel out. We can now solve for n. 

Now we have to plug in this value of n into any of our equations to find the value of m. 

Let's use the second equation.

Substitution needs to be used in order to solve this system of equations. From the second equation we know that ,

Substitute that into the first equation and solve.

You get

From there solve for y using the second equation.