PSAT Math : Equations / Inequalities

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #31 : Equations / Solution Sets

If  and , what is the value of ?

Possible Answers:

Correct answer:

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).

   

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

Solve for .

Possible Answers:

Correct answer:

Explanation:

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

Example Question #1 : Systems Of Equations

Solve for  in the system of equations:

Possible Answers:

The system has no solution

Correct answer:

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 #31 : Solving Equations

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

Line 1 : 

Line 2 : 

Possible Answers:

Correct answer:

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 #291 : Equations / Inequalities

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

Possible Answers:

Correct answer:

Explanation:

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 : Linear Equations With Whole Numbers

What is the solution of  for the systems of equations?

Possible Answers:

Correct answer:

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

What is the solution of  that satisfies both equations?

Possible Answers:

Correct answer:

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 #2 : Linear Equations With Whole Numbers

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

Possible Answers:

Correct answer:

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 : Systems Of Equations

What is the solution of  for the two systems?

Possible Answers:

Correct answer:

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 #34 : Equations / Solution Sets

Find the solutions for the following set of equations:

 

Possible Answers:

Correct answer:

Explanation:

If we multiply both sides of our bottom equation by , we get . We can now add our two equations, and eliminate , leaving only one variable. When we add the equations, we get . Therefore, . Finally, we go back to either of our equations, and plug in  so we can solve for .

 

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