# Algebra II : Solving Equations

## Example Questions

### Example Question #21 : Solving Equations

A large water tank has a water pipe that can be used to fill the tank in forty-five minutes. It has a drain that can empty the tank in one hour and twenty minutes.

One day, someone left the drain open when filling the tank. The tank was completely full by the time someone realized the error. Which of the following comes closest to the amount of time it took to fill the tank?

Explanation:

Work problems can be solved by looking at them as rate problems. Therefore, we can look at this problem in terms of tanks per minute, rather than minutes per tank. Let  be the number of minutes it took to fill the tank.

The pipe filled the tank at a rate of 45 minutes per tank, or  tank per minute; over a period of  minutes, it filled  tank.

The drain emptied the tank at a rate of 80 minutes per tank, so we can see this as a drain of  tank per minute. We can look at draining as "filling negative tanks" -  tank per minute; over a period of  minutes, it "filled"  tank.

Since their work adds up to one tank filled, We can set up, and solve for  in, the equation:

Using decimal approximations:

minutes, or 1 hour 43 minutes.

Of the given choices, 1 hour 45 minutes is closest.

### Example Question #501 : Basic Single Variable Algebra

Solve for x:

Explanation:

In order to solve for x, first subtract 4 from both sides of the equation:

Then, multiply both sides of the equation by 6:

Solve for :

Explanation:

### Example Question #24 : Solving Equations

Solve this system of equations:

Explanation:

Solve both equations for y:

Set them equal to each other and solve for :

Plug x back into either -equation to get

.

### Example Question #25 : Solving Equations

Solve for :

Explanation:

To solve for x we want to isolate x on one side of the equation and all other numbers on the other side. To do this we start with adding 5 to both sides.

Now we divide by 5 to solve for x.

### Example Question #26 : Solving Equations

Solve for :

Explanation:

In order to solve for , first add 9 to both sides of the equation:

Then, subtract  from both sides of the equation:

Finally, divide both sides of the equation by 4:

### Example Question #21 : Solving Equations

Solve for :

Explanation:

Add 6 to both sides of the equation:

Then divide both sides of the equation by 7:

### Example Question #24 : Solving Equations

Solve for .

Explanation:

This problem requires simplification, knowledge of order of operations, and ability to isolate variables and solve for .

Our first step is to simplify the left side of the equation. Use the distributive property to simplify

Distribute the   (Don't forget the negative!!)

Combine like terms

Now we have....

Combine like terms, simplify, and solve for

Subtract  from both sides

Subtract  from both sides

Divide by  on both sides

### Example Question #29 : Solving Equations

Solve for :

Explanation:

In order to solve this equation, we need to move all constants to one side and everything that has an  to the other side.

which becomes

Moving the  to the left side results in

or

Dividing each side by 13 gives us

### Example Question #22 : Solving Equations

Solve for  and :

Explanation:

There are two ways to solve this:

-The 1st equation can be mutliplied by  while the 2nd equation can be multiplied by  and added to the 1st equation to make it a single variable equation where

.

This can be plugged into either equation to get

or

-The 2nd equation can be simplified to,

.

This value for  can then be substituted into the first equation to make the equation single variable in .

Solving, gives , which can be plugged into either original equation to get