The most simple method for solving systems of equations is to transform one of the equations so it allows for the canceling out of a variable. In this case, we can multiply  by  to get .

 Then, we can add to this equation to yield , so .

We can plug that value into either of the original equations; for example, .

So,  as well.

By solving one equation for , and replacing  in the other equation with that expression, you generate an equation of only 1 variable which can be readily solved.

The number of pounds of coffee beans totals 50, so one of the equations would be

.

The total price of the Kona beans, is its unit price, $24 per pound, multiplied by its quantity,  pounds. This is  dollars. Similarly, the total price of the Ethiopian delight beans is  dollars, and the price of the mixture is  dollars. Add the prices of the Kona and Ethiopian Delight beans to get the price of the mixture:

We are trying to solve for  in the system

Multiply the second equation by , then add to the first:

The mixture includes 15 pounds of Kona beans.

To solve this problem, you must first solve the system of equations for both  and , then plug the values of  and  into the final equation.  

In order to solve a system of equations, you must add the equations in a way that gets rid of one of the variables so you can solve for one variable, then for the other. One example of how to do so is as follows:

Take the equations. Multiply the first equation by two so that there is  (this will cancel out the  in the second equation).

Add the equations:

Find the sum (notice that the variable  has disappeared entirely):

Solve for .

Plug this value of  back into one of the original equations to solve for :

Now, plug the values of  and  into the final expression:

The answer is . 

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

In the second equation, you can substitute  for  from the first.

Now, substitute 2 for  in the first equation:

The solution is 

To find the point where these two lines intersect, set the equations equal to each other, such that  is substituted with the  side of the second equation. Solving this new equation for  will give the -coordinate of the point of intersection.

Subtract from both sides.

Divide both sides by 2.

Now substitute  into either equation to find the -coordinate of the point of intersection.

With both coordinates, we know the point of intersection is . One can plug in  for  and  for  in both equations to verify that this is correct.

Add the equations together.

Put the terms together to see that .

Substitute this value into one of the original equaitons and solve for .

Now we know that , thus we can find the sum of and .

We add the two systems of equations:

For the Left Hand Side:

For the Right Hand Side:

So our resulting equation is:

Divide both sides by 10:

For the Left Hand Side:

For the Right Hand Side:

Our result is:

Reduce the second system by dividing by 3.

Second Equation:

     We this by 3.

Then we subtract the first equation from our new equation.

First Equation:

First Equation - Second Equation:

Left Hand Side:

Right Hand Side:

Our result is: