GMAT Math : Linear Equations, Two Unknowns
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Example Questions
Example Question #1 :Linear Equations, Two Unknowns
What is the value of z?
年代tatement 1:
年代tatement 2:
EACH statement ALONE is sufficient.
年代tatement 1 ALONE is sufficient, but statement 2 is not sufficient.
年代tatement 2 ALONE is sufficient, but statement 1 is not sufficient.
年代tatements 1 and 2 TOGETHER are NOT sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
年代tatements 1 and 2 TOGETHER are NOT sufficient.
To solve for three variables, you must have three equations. Statements 1 and 2 together only give two equations, so the statements together are not sufficient.
Example Question #2 :Dsq: Solving Linear Equations With Two Unknowns
Is the equation linear?
年代tatement 1:
年代tatement 2:
年代tatement 2 ALONE is sufficient, but statement 1 is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
年代tatement 1 ALONE is sufficient, but statement 2 is not sufficient.
年代tatements 1 and 2 TOGETHER are NOT sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
If we only look at statement 1, we might think the equation is not linear because of the
Example Question #3 :Dsq: Solving Linear Equations With Two Unknowns
Data sufficiency question- do not actually solve the question
年代olve for
1.
2.
Both statements taken together are sufficient to answer the question, but neither question alone is sufficient
年代tatement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question
Each statement alone is sufficient
年代tatements 1 and 2 together are not sufficient, and additional data is needed to answer the question
年代tatement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question
Each statement alone is sufficient
When solving an equation with 2 variables, a second equation or the solution of 1 variable is necessary to solve.
Example Question #4 :Dsq: Solving Linear Equations With Two Unknowns
Data Sufficiency Question
年代olve for
1.
2. Both
statements 1 and 2 together are not sufficient, and additional data is needed to answer the question
statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
each statement alone is sufficient
each statement alone is sufficient
Using statement 1 we can set up a series of equations and solve for both
Additionally, the information in statement 2 indicates that there is only one possible solution that satisfies the requirement that both are positive integers.
Example Question #1 :Linear Equations, Two Unknowns
Given that both
年代tatement 1:
年代tatement 2:
年代tatement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
年代tatement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
年代tatement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
If the slopes of the lines are not equal, then the lines intersect at one solution. If the slopes are equal, then there are two possibilties: either they do not intersect or they are the same line. Write each equation in slope-intercept form:
The slopes of these lines are
If Statement 1 is true, then we can rewrite the first slope as
Example Question #6 :Dsq: Solving Linear Equations With Two Unknowns
How many solutions does this system of equations have: one, none, or infinitely many?
年代tatement 1:
年代tatement 2:
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
年代tatement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
年代tatement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
If the slopes of the lines are not equal, then the lines intersect at one solution; if they are equal, then they do not intersect, or the lines are the same line. Write each equation in slope-intercept form,
The slopes of the lines are
We need to know both
年代et
The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.
Example Question #3 :Dsq: Solving Linear Equations With Two Unknowns
年代olve the following for x:
4x+7y = 169
1. x > y
2. x - y = 12
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
Each statement alone is sufficient.
年代tatement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.
年代tatements 1 and 2 together are not sufficient.
年代tatement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
年代tatement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.
To solve with 2 unknowns, we must create a system of equations with at least 2 equations. Using statement 2 as a second equation we can easily get our answer. Solve statement 2 for x or y, and plug in for the corresponding variable in the equation given by the problem.
年代o, solving statement 2 for x, we get x=12+y. Replacing x in the equation from the problem, we get 4(12+y) + 7y=169. We can distribute the 4, and combine terms to find 48+11y=169. Subtract 48 from both sides, we get 11y=121. So y=11. Reusing either equation and plugging in our y value gives our x value. So x - 11=12, or x=23. This shows that x > y, and statement 1 is true. But even though it's true, it is completely unneccessary information. Therefore the answer is that we only need the information from statement 2, and statement 1 is not needed.
Example Question #8 :Dsq: Solving Linear Equations With Two Unknowns
Given that
年代tatement 1:
年代tatement 2:
年代tatement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
年代tatement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
年代olve for
年代tatement 1:
年代tatement 2:
From either statement alone, it can be deduced that
Example Question #2 :Linear Equations, Two Unknowns
Data Sufficiency Question
年代olve for
1.
2.
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
年代tatements 1 and 2 together are not sufficient, and additional data is needed to answer the question
年代tatement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
Each statement alone is sufficient
年代tatement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Each statement alone is sufficient
In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, either of the statements allow the problem to be solved.
Example Question #10 :Dsq: Solving Linear Equations With Two Unknowns
Data Sufficiency Question
年代olve for
1.
2.
年代tatement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Each statement alone is sufficient
年代tatement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
年代tatements 1 and 2 together are not sufficient, and additional data is needed to answer the question
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, three equations are needed and both statements are required to solve the problem.
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