# PSAT Math : How to find the solution for a system of equations

## Example Questions

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### Example Question #1 : How To Find The Solution For A System Of Equations

Solve the system for  and .

Explanation:

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.

### Example Question #2 : How To Find The Solution For A System Of Equations

What is the solution to the following system of equations:

Explanation:

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.

### Example Question #1 : How To Find The Solution For A System Of Equations

Jeff, the barista at Moonbucks Coffee, is having a problem. He needs to make fifty pounds of Premium Blend coffee by mixing together some Kona beans, which cost $24 per pound, with some Ethiopian Delight beans, which cost$10 per pound. The Premium Blend coffee will cost $14.20 per pound. Also, the coffee will sell for the same price mixed as it would separately. How many pounds of Kona beans will be in the mixture? Possible Answers: Correct answer: Explanation: 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.

### Example Question #2 : How To Find The Solution For A System Of Equations

If  and , what is the value of ?

Explanation:

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).

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:

### Example Question #1 : How To Find The Solution To An Inequality With Division

Solve for .

Explanation:

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

### Example Question #111 : Algebra

Solve for  in the system of equations:

The system has no solution

Explanation:

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

Now, substitute 2 for  in the first equation:

The solution is

### Example Question #4 : How To Find The Solution For A System Of Equations

Without drawing a graph of either equation, find the point where the two lines intersect.

Line 1 :

Line 2 :

Explanation:

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.

### Example Question #22 : How To Find The Solution For A System Of Equations

What is the sum of and for the following system of equations?

Explanation:

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 .

### Example Question #241 : Equations / Inequalities

What is the solution of  for the systems of equations?

Explanation:

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:

### Example Question #2 : Linear Equations With Whole Numbers

What is the solution of  that satisfies both equations?

Explanation:

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:

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