In order to solve this system of equations, it will probably be best to use the elimination method, in which we add or subtract two equations. In general, if , and , then . In other words, we can combine equations.

We need to solve the following equations:

We want to look for a way to combine these two equations that will eliminate at least one variable. If, for example, we multiplied the first equation by  and then added this to the second equation, we could get rid of the  terms. Let's multiply the first equation by . Remember, we must multiply both sides.

Distribute the .

We can now add to the equation . The sum of the left sides of each equation will equal the sum of the right sides.

We can simplify both sides. Notice that the  terms on the left disappear, leaving only  terms.

Divide both sides by 3.

.

Now that we have , we can go back to either of our original two equations, substitute the value of  we obtained, and solve for . Let's use the equation .

Substitute  for .

Add  to both sides. We will need to rewrite 4 with a denominator of 3.

Divide by 2 on both sides. When dividing a fraction, we simply multiply by its reciprocal.

Point , which is the intersection of the lines, must have the coordinates .

The question ultimately asks for the slope of the line passing through  and the origin. We can use the equation of the slope between two points with coordinates and .

slope =

The answer is .

Finding the intersection of two lines is the same as solving simultaneous equations.  Here, both equations are already solved for , so the right sides of both equations are set equal to each other.  Solving for  results in a value of 11.  Setting  equal to 11 in either of the original equations gives a  value of 27.

Let x equal the number of hamburgers and y equal the number of hotdogs.

Solve for y in the first equation.

Substitute that solution into the second equation.

Let Amy's current age be equal to A and Jackie's age be equal to J.

Therefore, and

Substitute the first equation into the second.

Therefore, today Jackie is 6 years old.

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 .