# GMAT Math : Solving linear equations with one unknown

## Example Questions

### Example Question #11 : Solving Linear Equations With One Unknown

Which of the following equations has the set of all real numbers as its solution set?

Explanation:

The right side of each equation is , which simplifies by way of distribution to

If the left side of the equation simplifies to an identical expression, the equation has all real numbers as its solutions.

We test the left side of each equation:

\

Of the given choices,

can be rewritten as

which is an identity and has the set of all real numbers as its solution set.

### Example Question #12 : Solving Linear Equations With One Unknown

Consider the incomplete equation

What number replaces the box in order to form an equation with no solution?

None of the other choices yields a correct answer.

None of the other choices yields a correct answer.

Explanation:

Set  to be the number that replaces the box.

Simplify first:

Now solve for  in terms of :

The only possible value of  that might preclude the existence of a solution is , since it makes the denominator 0. However, let us test this value in the original equation:

As it turns out, replacing the box with 15 yields an identity, not a contradiction, so the solution set is the set of all real numbers. There is no number that fits the description.

### Example Question #13 : Solving Linear Equations With One Unknown

Consider the incomplete equation

Which of the following numbers can replace the box to form an equation whose one and only solution is 2?

None of the other responses gives a correct answer.

Explanation:

Rewrite this equation as

If 2 is a solution of the equation, then we can substitute 2 for  to make a true arithmetic equation. Replace  with 2 and solve for :

This number replaces the box in order to form the equation

### Example Question #14 : Solving Linear Equations With One Unknown

Solve the following equation for

Explanation:

We proceed as follows

(Start)

(Multiply both sides by 4. Remember to distribute the 4 to both summands on both sides.)

(Subtract 28 from both sides)

(Subtract 4x from both sides)

(Divide both sides by 2)

### Example Question #15 : Solving Linear Equations With One Unknown

Define a function  as follows:

If , evaluate .

Explanation:

Since , we can plug N in for x and 47 in for f(N) to get the following equation,

From here, we want to solve for N therefore we must isolate N on one side of the equation and all other numbers on the other side.

### Example Question #16 : Solving Linear Equations With One Unknown

Lisa went to a bargain bookstore where books were sold for  dollars and magazines for  dollars. After buying six books and four magazines, she only spent $30.00. How much did the books and magazines cost? Possible Answers: Books:$1.80; Magazines: $3.80 Books:$3.80; Magazines: $1.80 Books:$4.20; Magazines: $2.20 Books:$3.80; Magazines: $5.80 Correct answer: Books:$3.80; Magazines: $1.80 Explanation: We must first write out the equation to this problem. • If we set to be the cost of books then we can set to represent the cost of magazines. • She purchased books which means she spent on books. • She also bought magazines which means she spent on magazines. • The total spent was so when these two values are added together they must equal : We can now solve for Remember: the cost of magazines was so we must plug in the value to find our answer. So, our answer is that books cost$ and magazines cost \$

### Example Question #17 : Solving Linear Equations With One Unknown

Students at a local college decide to make and sell t-shirts in order to raise money for their activities. They will pay the manufacturer  per t-shirt made and a fixed fee of . If they sell each t-shirt for , how many t-shirts would they have to make and sell to raise ?

Explanation:

We set up the following equation:

The students will need to make and sell 180 t-shirts in order to raise .

### Example Question #18 : Solving Linear Equations With One Unknown

Solve for :

Explanation:

In order to solve for , we must isolate  on one side of the equation:

### Example Question #19 : Solving Linear Equations With One Unknown

What is the value of  in the following equation when ?

Explanation:

When solving algebraic equations with one unknown, it is often easier to rearrange the equation first so that you have the unknown variable isolated.

So this:

becomes this when we subtract  from both sides:

Then, divide both sides by  to get  all by itself:

To finish, simply plug in  for  and simplify.

Solve for