# SAT II Math II : Solving Equations

## Example Questions

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### Example Question #1 : Finding Roots

Solve for :

Explanation:

To solve for , you need to isolate it to one side of the equation. You can subtract the  from the right to the left. Then you can add the 6 from the right to the left:

Next, you can factor out this quadratic equation to solve for . You need to determine which factors of 8 add up to negative 6:

Finally, you set each binomial equal to 0 and solve for :

### Example Question #2 : Single Variable Algebra

Give the solution set of the equation  .

Explanation:

Use the quadratic formula with :

### Example Question #7 : Single Variable Algebra

A large tub has two faucets. The hot water faucet, if turned all the way up, can fill the tub in 15 minutes; the cold water faucet can do the same in 9 minutes. Which of the following responses is closest to the time it takes to fill the tub if both faucets are turned all the way up?

Explanation:

Work problems can be solved by looking at them as rate problems.

The hot faucet can fill up the tub at a rate of 15 minutes per tub, or  tub per minute. The cold faucet, similarly, can fill up the tub at a rate of 9 minutes per tub, or  tub per minute.

Suppose the tub fills up in  minutes. Then, at the end of this time, the hot faucet has filled up  tub, and the cold faucet has filled up  tub, for a total of one tub. We can set up this equation and solve for :

Of the five choices, 6 minutes comes closest.

### Example Question #11 : Single Variable Algebra

Solve the equation for y

Explanation:

First subtract 27 from both sides of the equation

Add 5z to both sides of the equation

Lastly, divide both sides by 5 to get the y by itself

### Example Question #2 : Solving Equations

Solve the equation:

Explanation:

To isolate the x-variable, we can multiply both sides by the least common denominator.

The least common denominator is .  This will eliminate the fractions.

Subtract 4 on both sides.

Divide by 24 on both sides.

### Example Question #1 : Solving Equations

Solve the equation:

Explanation:

Find the least common denominator of both sides of the equation, and multiply it on both sides.

The LCD is 60.

Combine like-terms on the left.

Divide by 5 on both sides.

### Example Question #11 : Single Variable Algebra

Solve the equation:

Explanation:

Distribute the eight through both terms of the binomial.

Divide by 9 on both sides.

### Example Question #1 : Solving Equations

Solve the equation:

Explanation:

Subtract  from both sides.

### Example Question #1 : Solving Equations

Solve

Explanation:

First, we want to get everything inside the square roots, so we distribute the :

Now we can clear our the square roots by squaring each side:

Now we can simplify by moving everything to one side of the equation:

Factoring will give us:

### Example Question #11 : Single Variable Algebra

Solve

No solution.

No solution.

Explanation:

Begin by gathering all the constants to one side of the equation:

Now multiply by :

And finally, square each side:

This might look all fine and dandy, but let's check our solution by plugging it in to the original equation:

So our solution is invalid, and the problem doesn't have a solution.

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