Let "a number" be represented by the variable .

"The quotient of a number and nine" is  divided by 9, or

.

"Eight less than" means we subtract 8 from this (we do not subtract this from eight):

Let " a number" be expressed as the variable .

"Two-thirds of a number" is the product of the two, and is expressed as 

"Increased by twelve" means 12 is added to the expression. The correct response is therefore

"A number" can be represented by the variable .

"The sum of a number and three" is the result of an addition and can be represented as

"Seven divided into" this expression is the expression  divided by seven, so this expression can be represented as

.

The first step is to get  by itself and it is also helpful to do away with negative values. When manipulating an equation perform the opposite algebraic operation to keep the equivalency of the equation true.

So you must add both  and  to each side resulting in 

.  

The final step is dividing both sides by  which gives you an answer of 

.

Begin by distributing the  to its entire group:

Next, group the like terms:

Finally, combine the like terms:

When you are asked to find the sum that means we are going to add. Because the phrase says "then divide" we need to put the addition problem in parentheses because of our order of operations. Then we can divide. 

Remember, order of operations is PEMDAS= parentheses, exponents, multiplication/division, addition/subtraction.  

When you are asked to find the product that means we are going to multiply. Because we are multiplying and dividing in this question, we do not need to use parentheses because with multiplication and division you work the problem out from left to right. So first we have the multiplication problem, then we divide. 

Remember, order of operations is PEMDAS= parentheses, exponents, multiplication/division, addition/subtraction.  

Quotient means the answer to a division problem. Because we want to divide  by the answer of  we need to put that in parentheses so that it's done first. 

Remember, order of operations is PEMDAS= parentheses, exponents, multiplication/division, addition/subtraction.  

Because we are multiplying by the sum, which is the answer to an addition problem, we first need to add so we need to put the addition problem in parentheses, then we can do the multiplication. 

Remember, order of operations is PEMDAS= parentheses, exponents, multiplication/division, addition/subtraction.  

Because we are dividing by the difference, or answer, from the subtraction problem, we first need to subtract. In order for the subtraction problem to go first, we need to put that in parentheses. 

Remember, order of operations is PEMDAS= parentheses, exponents, multiplication/division, addition/subtraction.