Linear Equations, One Unknown - GMAT Quantitative Reasoning
Card 0 of 16
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.
Barry left home at
(1) Barry drove for
(2) Barry stopped at a gas station halfway there and drove an additional
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.
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.
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Find 
, a linear equation passing through the point 
.
I) 
is parallel to 
which is given by:
II) 
has a slope of 
.
Find
I)
II)
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.
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.
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Given:
Find 
.
I) 
.
II) 
and passes through 
.
Given:
Find
I)
II)
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 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.
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Given:
Find 
.
I) 
.
II) 
crosses the 
-axis at 
.
Given:
Find
I)
II)
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 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.
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What is 
?
(1) 
(2) 
What is
(1)
(2)
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 answer the question we must know the absolute value of
Statement 1 tells us the absolute value of
Statement 2 also tells us that the absolute value of
Therefore, the final answer is each statement alone is sufficient.
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What is 
?
(1) 
(2) 
What is
(1)
(2)
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 be able to answer the question, we must have a definitive value for
Statement 1 tells us that
Statement 2 tells us that the cube of
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What is 
?
(1) 
(2) 
What is
(1)
(2)
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 answer the question, we should be able to find a single value for
Statement 1 gives us two possible values for
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
Hence, statements 1 and 2 taken together are not sufficient.
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, what is 
if 
is an integer?
(1) 
(2) 
(1)
(2)
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.
To begin with, we should see that information about unknowns
Statement 1 gives us information about the upper bound for
Statement 2 gives us information about the lower bound for
Taking these statements together we get that
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What is 
?
(1) 
is positive
(2) 
(1)
(2)
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.
Firstly, we should try to simplify the equation, to see solutions for
So we have to solutions for the equation and we need to know what
Statement 1 tells us that
Statement 2 tells us that
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. What is 
?
(1) 
(2) 
is an integer
(1)
(2)
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 
.
First, we should try to simplify the quadratic equation, and we get
Statement 1 tells us that
Statement 2 tells us that
Statements 1 and 2 together are still insufficient, since none can help us find a single value for
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. What is 
?
(1) 
(2) 
(1)
(2)
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.
Firstly, we should try to find a simplified equation to better see the possible values for
Statement 1 tells us that the square of
Statement 2 tells us that the absolute value of
Note that it is possible to answer with either statement only because
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What is 
if, 
?
(1) 
(2) 
What is
(1)
(2)
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 begin with this problem, we should try to solve
Statement 1 gives us an other unknown for the value of
Statement 2 gives an absolute value for this unknown
Taken together these statements, allow us to see that
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. What is 
?
(1) 
(2) 
(1)
(2)
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 approach this problem, we should firstly set up two possible equations for the value of
Statement 1 tells us that
Statement 2 tells us that
Because there is the absolute value we get two equations:
or
Therefore, statement 2 alone is sufficient
To conclude, each statement alone is sufficient.
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What is 
?
(1) 
(2) 
What is
(1)
(2)
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 answer this question, we must find a single value for
Statement 1 gives us an equation with two possible solutions for
Statemnt 2 alone is also insufficient, because it gives us the same possible values for
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.
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, What is 
?
(1) 
(2) 
(1)
(2)
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.
To find a value for
Statement 1 has two unknowns therefore we need another different equation with
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
Both statements together are sufficient.
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and 
is different than 
. What is 
?
(1) 
(2) 
is not an integer.
(1)
(2)
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.
Firstly we should try to see what are the possible values for
either
This allows us to find two values for
Statement 1 tells us that
Statement 2 tells us that
Statement 1 alone is sufficient.
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