DSQ: Solving linear equations with one unknown - GMAT Quantitative

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.

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.

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.

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.

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.

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.

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.

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.

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 .

Firstly, we should try to find a simplified equation to better see the possible values for . We get . We can see that can either be or .

Statement 1 tells us that the square of is . It follows that must be , therefore, this statement is sufficient.

Statement 2 tells us that the absolute value of is , therefore must be and therefore the statement is also sufficient alone.

Note that it is possible to answer with either statement only because can either be or . If could have been or than statements 1 and 2 would have been insufficient.

To begin with this problem, we should try to solve . It gives us two sets of equations for us to find values for ; and . Solving gives us two possible values for , and . Let's see how can the statements help us determine the value of .

Statement 1 gives us an other unknown for the value of . Therefore, this statement is insufficient.

Statement 2 gives an absolute value for this unknown . But we don't know what other values is equal to.

Taken together these statements, allow us to see that must be and therefore are sufficient to answer the question.

To approach this problem, we should firstly set up two possible equations for the value of ; either or .

Statement 1 tells us that is in fact zero. Than the equations return a single value for , therefore statement 1 alone is sufficient.

Statement 2 tells us that , if plug in this value for , we get that:

Because there is the absolute value we get two equations:

or

must be 8. Plugging in the value for in our first equation gives us no solution because both sides are of equal value and we end up with .

Therefore, statement 2 alone is sufficient

To conclude, each statement alone is sufficient.

To answer this question, we must find a single value for .

Statement 1 gives us an equation with two possible solutions for . Therefore, statement 1 alone is not sufficient, since can either be or

Statemnt 2 alone is also insufficient, because it gives us the same possible values for as the equation in statement 1.

When two statements give us the same the information the answer is either both statements together are sufficient or statements 1 and 2 together are not sufficient. Here neither statement allowed us to answer, it follows that statements 1 and 2 together are not sufficient.

To find a value for , we should be able to get a value for .

Statement 1 has two unknowns therefore we need another different equation with to be able to find values for these unknowns.

Statement 2 alone is also insufficient because just as statement 1 has two variables and therefore we need more information to solve it.

Taking together these equations, by adding both sides we get and from there we can find a single value for .

Both statements together are sufficient.

Firstly we should try to see what are the possible values for , by solving the equations given by the absolute value:

either or .

This allows us to find two values for which are and , let's see how the statements can help us determine a single value for .

Statement 1 tells us that must be greater than one. Only one of our solutions for is greater than one. Therefore, statement 1 alone is sufficient.

Statement 2 tells us that is not an integer, however both solutions are not integer values and therefore statement 2 doesn't help us find a single solution.

Statement 1 alone is sufficient.

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.

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.

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.

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.