# PSAT Math : Equations / Solution Sets

## Example Questions

### 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 #201 : Algebra

Solve for .

Explanation:

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

### Example Question #1 : Linear Systems With Two Variables

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 #3 : 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 #4 : 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 #261 : Algebra

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 #31 : Equations / Solution Sets

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:

### Example Question #4 : Creating Equations With Whole Numbers

What is the solution of  for the two systems of equations?

Explanation:

We first add both systems of equations.

Left Hand Side:

Right Hand Side:

Our resulting equation is:

We divide both sides by 3.

Left Hand Side:

Right Hand Side:

Our resulting equation is:

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

What is the solution of  for the two systems?

Explanation:

We first multiply the second equation by 4.

So our resulting equation is:

Then we subtract the first equation from the second new equation.

Left Hand Side:

Right Hand Side:

Resulting Equation:

We divide both sides by -15

Left Hand Side:

Right Hand Side:

Our result is:

### Example Question #31 : Equations / Inequalities

Find the solutions for the following set of equations: