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 multiplybyto get.

Then, we can addto 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 replacingin the other equation with that expression, you generate an equation of only 1 variable which can be readily solved.

的麻木er 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 isdollars. Similarly, the total price of the Ethiopian delight beans isdollars, and the price of the mixture isdollars. Add the prices of the Kona and Ethiopian Delight beans to get the price of the mixture:

We are trying to solve forin 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 bothand, then plug the values ofandinto 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 thein the second equation).

Add the equations:

Find the sum (notice that the variablehas disappeared entirely):

Solve for.

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

现在,塞薇尔ues ofandinto the final expression:

The answer is.

For the second equation, solve forin terms of.

Plug this value of y into the first equation.

In the second equation, you can substituteforfrom the first.

Now, substitute 2 forin the first equation:

The solution is

To find the point where these two lines intersect, set the equations equal to each other, such thatis substituted with theside of the second equation. Solving this new equation forwill give the-coordinate of the point of intersection.

Subtractfrom both sides.

Divide both sides by 2.

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

With both coordinates, we know the point of intersection is. One can plug inforandforin 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 ofand.

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: