### All PSAT Math Resources

## Example Questions

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

Solve the system for and .

**Possible Answers:**

**Correct answer:**

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

What is the solution to the following system of equations:

**Possible Answers:**

**Correct answer:**

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 #3 : 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:**

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

If and , what is the value of ?

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #2 : Inequalities

Solve for .

**Possible Answers:**

**Correct answer:**

For the second equation, solve for in terms of .

Plug this value of y into the first equation.

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

Solve for in the system of equations:

**Possible Answers:**

The system has no solution

**Correct answer:**

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

Now, substitute 2 for in the first equation:

The solution is

### Example Question #6 : 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 :

**Possible Answers:**

**Correct answer:**

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 #7 : How To Find The Solution For A System Of Equations

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

**Possible Answers:**

**Correct answer:**

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 .

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

What is the solution of for the systems of equations?

**Possible Answers:**

**Correct answer:**

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 #31 : How To Find The Solution For A System Of Equations

What is the solution of that satisfies both equations?

**Possible Answers:**

**Correct answer:**

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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