# GMAT Math : DSQ: Solving linear equations with one unknown

## Example Questions

### Example Question #1 : Linear Equations, One Unknown

Barry left home at  am to go to his friend Harry's house. How many miles away is Harry's house from Barry's house?

(1) Barry drove for  minutes at  mph

(2) Barry stopped at a gas station halfway there and drove an additional  miles before reaching Harry's house.

E: Statements (1) and (2) TOGETHER are not sufficient

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

D: EACH statement ALONE is sufficient

B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

D: EACH statement ALONE is sufficient

Explanation:

Statement 1 gives us the time and the speed so we can derive the distance:

.

Therefore,

Statement 2 tells us that the gas station is halfway betweent the two houses so:

distance between Barry's house and gas station = distance between Harry's house and gas station = 0.5 distance between the two houses

Furthermore, we learn that the distance between the gas station and Harry's house is 20 miles.

Therefore:

distance between Harry's house and gas station = 20 miles = 0.5 distance between the two houses

So the distance = 2 x 20 = 40 miles

So the correct answer is D; each statement alone is sufficient.

### Example Question #2 : Dsq: Solving Linear Equations With One Unknown

Find , a linear equation passing through the point .

I)  is parallel to  which is given by:

II)  has a slope of .

Statement 2 is sufficient to solve the question, but statement 1 is not sufficient to solve the question.

Each statement alone is enough to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Statement 1 is sufficient to solve the question, but statement 2 is not sufficient to solve the question.

Both statements taken together are sufficient to solve the question.

Each statement alone is enough to solve the question.

Explanation:

We are given a point and two clues.

Both I and II give us the slope of f(x). It must be 4 because we are told so in II. This holds true from statement I since it must be parallel to g(x), which has a slope of four.

With a slope and a point we can find the equation of f(x) using the point slope form,

.

Therefore either statement alone is enough.

### Example Question #3 : Dsq: Solving Linear Equations With One Unknown

Given:

Find .

I) .

II)  and passes through .

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Explanation:

To find b, we need a point on k(t).

I) Gives us that point.

II) Gives us some details about a parallel line, which is cool and all, but it doesn't help us find b.

So statement I alone is sufficient to answer the question.

### Example Question #4 : Dsq: Solving Linear Equations With One Unknown

Given:

Find .

I) .

II)  crosses the -axis at .

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Either statement is sufficient to answer the question.

Explanation:

To find m, we need a point on the line.

Both I and II give us points, so we can use either of them to solve for m.

### Example Question #5 : Dsq: Solving Linear Equations With One Unknown

What is ?

(1)

(2)

Statement 1 alone is sufficient.

Statements 1 and 2 together are sufficient.

Both statements together are sufficient.

Statement 2 alone is sufficient.

Each statement alone is sufficient.

Each statement alone is sufficient.

Explanation:

To answer the question we must know the absolute value of .

Statement 1 tells us the absolute value of , indeed, it is .

Statement 2 also tells us that the absolute value of  is 5, since .

Therefore, the final answer is each statement alone is sufficient.

### Example Question #6 : Dsq: Solving Linear Equations With One Unknown

What is ?

(1)

(2)

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Statements 1 and 2 together are sufficient.

Each statement alone is sufficient.

Both statements together are sufficient.

Statement 2 alone is sufficient.

Explanation:

To be able to answer the question, we must have a definitive value for

Statement 1 tells us that  is , in other words  could be two values  or . This statement gives us two possibilities for  and is therefore insufficient.

Statement 2 tells us that the cube of  is , therefore  must be . This statement gives a single possible value for  and therefore is, alone, sufficient.

### Example Question #1 : Dsq: Solving Linear Equations With One Unknown

What is ?

(1)

(2)

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Statements 1 and 2 together are not sufficient.

Both statements together are sufficient.

Each statement alone is sufficient.

Statements 1 and 2 together are not sufficient.

Explanation:

To answer the question, we should be able to find a single value for

Statement 1 gives us two possible values for . Indeed,  or . Hence, the information provided doesn't allow us to find the answer to the problem.

Statement 2 although a complicated equation to calculate, won't prove useful because the power is an even number and therefore, the equation will also have two solutions.

Both statements together are not sufficient because they both give us the value of , which is not sufficient.

Hence, statements 1 and 2 taken together are not sufficient.

### Example Question #8 : Dsq: Solving Linear Equations With One Unknown

, what is  if  is an integer?

(1)

(2)

Each statement alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Statement 2 alone is sufficient.

Statement 1 alone is sufficient.

Both statements together are sufficient.

Explanation:

To begin with, we should see that information about unknowns  or  would be useful to answer the problem. We already know that both these unknowns are integers.

Statement 1 gives us information about the upper bound for . However,  can still be an infinity of values, therefore this statement alone is insufficient.

Statement 2 gives us information about the lower bound for , just as statement 1, this statement alone doesn't allow us to find a single value for .

Taking these statements together we get that . Since  is an integer,  can only be . Both statements together are sufficient.

### Example Question #2 : Dsq: Solving Linear Equations With One Unknown

What is ?

(1)  is positive

(2)

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Each statement alone is sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Statement 2 alone is sufficient.

Explanation:

Firstly, we should try to simplify the equation, to see solutions for . We get . The best west way to simplify quadratic equations is to find the possible factors for the last term  in the general quadratic equation  and those two factors must add up to . Here for example,  and  add up to  and their products is .

So we have to solutions for the equation and we need to know what  we are looking for.

Statement 1 tells us that  is positive, however, the two possible solutions are positive and therefore, statement 1 doesn't help us find the correct solution for .

Statement 2 tells us that  is smaller than 3. Only one of our solutions is smaller than 3. Therefore statement 2 alone is sufficient.

### Example Question #10 : Dsq: Solving Linear Equations With One Unknown

. What is ?

(1)

(2)  is an integer

Statements 1 and 2 together are not sufficient.

Each statement alone is sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Explanation:

First, we should try to simplify the quadratic equation, and we get . This allows to see the two solutions for our equation.

Statement 1 tells us that  is between  and . But both possible solutions are in this interval. Therefore statement 1 alone is not sufficient.

Statement 2 tells us that  is an integer, which we already knew by reducing the equation. Therefore, this statements doesn't help us find a single value for

Statements 1 and 2 together are still insufficient, since none can help us find a single value for .

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