College Algebra : Equations with more than One Variable

Study concepts, example questions & explanations for College Algebra

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Example Questions

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Example Question #1 : Equations With More Than One Variable

Solve for X:

 

Possible Answers:

 

Correct answer:

Explanation:

In , if we're solving for x, we first need to get the "x" term isolated. We do this by subtracting 3 from both sides so:

becomes

Now we divide both sides by 8

Re-written the answer becomes

 

Example Question #2 : Equations With More Than One Variable

Larry has a handful of dimes and quarters. In total, he has 14 coins with a value of $2.60. How many of each coin does he have?

Possible Answers:

9 Dimes

5 Quarters

10 Dimes

4 Quarters

8 Dimes

6 Quarters

7 Dimes

7 Quarters

6 Dimes

8 Quarters

Correct answer:

6 Dimes

8 Quarters

Explanation:

Since this problems has 2 variables (D-dimes and Q-quarters) we need 2 equations. Because Larry has 14 coins, the first equation can be written as:

 

 

The value of those coins equals $2.60 or 260 cents. If Dimes are worth 10C and quarters are 25C, the next equation can be written as

 

To solve this write both equations on top of each other

 

 

 

Now we eliminate 1 variable by multiplying 1 equation by the lowest common denominator (as a negative) and adding the equations together.

  becomes

adding the equations

   

-----------------------------------

                 

 

now we solve for Q.

Since we know Q, now we plug it back in to an equation and find D

 

 

Larry has 6 dimes and 8 quarters

 

 

Example Question #3 : Equations With More Than One Variable

Solve for X and Y for the following pair of equations

Possible Answers:

Correct answer:

Explanation:

There are two ways to solve for x and y in a pair of equations. One way is to add the two equations together and eliminate one of the variables. It may be necessary to multiply one equation by a positive or negative number in order to cancel out one of the variables. The second way is to pick one of the equations and solve for one of the variables. Let's pick the top equation and solve for x:

Now substitute x on the other (bottom) equation:

Now, since we know the value of y, use either equation and "plug in" the value of y:

To check your answers, you can plug in both answers to either equation, and since they are equations, if both sides are equal, your answers are correct.

Example Question #4 : Equations With More Than One Variable

Solve for X and Y with the following set of equations:

    

Possible Answers:

Correct answer:

Explanation:

There are 2 ways to solve this set of equations. First, you can solve for one of the variables, then substitute that value for the variable in the other equation. The other way is to add the equations together (combine), and to do this, sometimes multiplying one of the equation by a positive or negative number is required. Let's see how we can do this:  

Remember the key is to add the two equations together and eliminate one of the variables:

   

Look at the x and y variables. Notice how if we added the two equations together, we can eliminate the x variable and we can thus solve for y:

So now, add the two equations together:

We got this buy adding 4y + 2y and 22+2. Now let's divide the equation by 6, so we can solve the value of y.

     so

       

Now plug the known value of y into either of the two equations. Let's use the top equation. Remember it doesn't matter which equation you pick to solve for x:

   

If you want to double-check your answers, just insert your values of x and y into the equation, and remember that since it's an equation, both sides will be equal.

 

Example Question #3 : Equations With More Than One Variable

Solve for X and Y for in the following pair of equations

Possible Answers:

Correct answer:

Explanation:

There are 2 ways to solve this set of equations. First, you can solve for one of the variables, then substitute that value for the variable in the other equation. The other way is to add the equations together (combine), and to do this, sometimes multiplying one of the equation by a positive or negative number is required. Let's see how we can do this:  

Remember the key is to add the two equations together and eliminate one of the variables.  

so let's look at the pair of equations. Both x's are positive, so we'd have to multiply one of the equations by -1 to eliminate the x variable. Let's look at the y. One is positive, and one is negative, so adding the equations as they are would eliminate y. So after you add the two equations together, you will get:

 

The y was eliminated as you can see, so now we can solve for x. Let's divide both sides by 2 to isolate the x variable:

Now that we have the value of x, all you have to do is plug the value of x into either of the two equations to solve for y. Let's pick the top equation:

Now subtract 4 from each side to isolate the y and solve for y:

If you want to check your answers, just plug the values you got for x and y into either of the pair of equations. Since they are equations, both sides will be equal if you found the correct values for x and y.

 

 

 

Example Question #6 : Equations With More Than One Variable

Solve for x and y in the following pair of equations

   

Possible Answers:

Correct answer:

Explanation:

There are 2 ways to solve this set of equations. First, you can solve for one of the variables, then substitute that value for the variable in the other equation. The other way is to add the equations together (combine), and to do this, sometimes multiplying one of the equation by a positive or negative number is required. Let's see how we can do this:  Remember the key is to add the two equations together and eliminate one of the variables. Let's look at the pair of equations. With either of the variables, since both x an y are positive and negative, you just have to multiply one of the equations in order to eliminate one of the variables. Let's eliminate y.  

   

    

In order to eliminate y, we will have to multiply the bottom equation by 2. When adding the two equations together, -4y and 4y will then cancel each other out, allowing you to solve for x.

So now this is the new pair of equations, where y is eliminated. Now just add the two equations together:

      

So the resulting equation after adding these equations together is:

Now divide both sides by -9 to solve for x:

Now just plug in the value of x into either of the two equations. Let's pick the top equation:

 

Now subtract six from both sides to isolate the y:

   

Now divide both sides by -4 to solve for y;

If you want to check your answers, just plug in your values of x and y into either of the equations. Since they are equations, if both sides are equal, you got the right answers.

 

Example Question #7 : Equations With More Than One Variable

Solve for x and y in the following pair of equations

Possible Answers:

Correct answer:

Explanation:

There are two ways to solve for x and y in a pair of equations. One way is to add the two equations together, eliminating one of the variables. This may require multiplying one of the equations by a positive or negative number. The other method is to pick one of the equations and solve for one of the variables. Then plug the value you found for the variable into the other equation and solve for a variable. Then you will plug in the solved value for that variable into either equation and solve for the other equation. Remember, you can solve for x and y using either of these methods. This time, let's solve for x then substitute that value into the other equation:

Let's solve for x using the bottom equation. It would be easier than to divide the top equation by 4 to solve for x: So let's subtract 3y from each side in order to solve for x:

   

Now just plug this into the other (top) equation to solve for y

   

Now subtract 64 from both sides to isolate y:

     

Now divide both sides by -11 to solve for y:

     

Now just plug in the known value of y into either of the equations in order to solve for x. Let's pick the top equation:

   

Now subtract three from each side:

             

Now divide both sides by four to solve for x:

If you want to check your answers, plug in both values of x and y into either equation, and since it's an equation, both sides will be equal if your answers are correct.

Example Question #8 : Equations With More Than One Variable

Solve for x and y in the following pair of equations

Possible Answers:

Correct answer:

Explanation:

There are two ways to solve for x and y in a pair of equations. The first method is to add the two equations together, eliminating a variable. This may require multiplying an equation by a positive or negative number in order to eliminate one of the variables. Since there are odd and even numbers in front of the variables in this case, it would be easier to use the second method, which is solving for a variable, then plugging in that value into the other equation. So let's pick the bottom equation to solve for x:

   

Subtract 2y from each side to isolate x:

    

       

Now divide both sides by 2 to solve for x:

     

Now plug in this value for x into the other (top) equation:

       

Subtract 54 from each side to isolate y:

       

Now divide both sides by -16 to solve for y:

     

Now plug in the value of y into either equation to solve for x:

Let's pick the bottom equation:

       

Now subtract six from both sides to isolate x:

                 

Now divide both sides by 2 to solve for x:

You can check your answers by plugging in your answers into either of the equations:  

Since they are equations, if both sides are equal, your answers are correct.

 

Example Question #9 : Equations With More Than One Variable

Solve for x and y in the following pair of equations

Possible Answers:

Correct answer:

Explanation:

There are 2 ways to solve this set of equations. First, you can solve for one of the variables, then substitute that value for the variable in the other equation. The other way is to add the equations together (combine), and to do this, sometimes multiplying one of the equation by a positive or negative number is required. Let's see how we can do this:  

Remember the key is to add the two equations together and eliminate one of the variables:

In this case, you can see how if you add the two equations together, the y variable is eliminated, since y and -y eliminate each other:

     

When you combine the two equations, you get:

         

Divide both sides by 16 to solve for x:

               

To solve for y, plug in the value of x into either equation.      

Let's  pick the top equation:

   

Now subtract 18 from each side to solve for y:

You can check your answers by plugging your answers into either of the two equations. Since it is an equation, both sides will be equal if your answers are correct.

 

Example Question #4 : Equations With More Than One Variable

Solve for x and y in the following pair of equations

Possible Answers:

Correct answer:

Explanation:

There are 2 ways to solve this set of equations. First, you can solve for one of the variables, then substitute that value for the variable in the other equation. The other way is to add the equations together (combine), and to do this, sometimes multiplying one of the equation by a positive or negative number is required. Let's see how we can do this:  

Remember the key is to add the two equations together and eliminate one of the variables.

In this case, you can see how you can eliminate the y variable by multiplying the bottom equation by 2 so that way 4y and -4y will cancel each other out:

   

so now adding the top equation and the bottom equation that was multiplied by 2 gives you:

     

Now divide both sides by 21 to solve for x:

   

Now pick either equation and plug in the value of x to solve for y:

Let's pick the top equation:

   

Now subtract 18 from each side to isolate y:

           

Now divide both sides by 4 to solve for y:

To check your answers, just plug in your answers into either equation, and since it is an equation, both sides will be equal if your answers are correct.

 

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