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?
Statement 1:
Statement 2:
Statement 1 ALONE is sufficient, but statement 2 is not sufficient.
Statements 1 and 2 TOGETHER are NOT sufficient.
EACH statement ALONE is sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement 2 ALONE is sufficient, but statement 1 is not sufficient.
Statements 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 : Linear Equations, Two Unknowns
Is the equation linear?
Statement 1:
Statement 2: 
Statements 1 and 2 TOGETHER are NOT sufficient.
Statement 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.
Statement 1 ALONE is sufficient, but statement 2 is 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 #1 : Linear Equations, Two Unknowns
Data sufficiency question- do not actually solve the question
Solve for 
1.
2.
Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question
Each statement alone is sufficient
Statement 2 alone is sufficient but statement 1 alone is not sufficient 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 question alone is sufficient
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 #3 : Dsq: Solving Linear Equations With Two Unknowns
Data Sufficiency Question
Solve for 
1.
2. Both 
statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
statements 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
each statement alone is sufficient
statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
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 #912 : Data Sufficiency Questions
Given that both 
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 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 #2 : Dsq: Solving Linear Equations With Two Unknowns
How many solutions does this system of equations have: one, none, or infinitely many?
Statement 1:
Statement 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.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 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 
Set 
The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.
Example Question #5 : Dsq: Solving Linear Equations With Two Unknowns
Solve the following for x:
4x+7y = 169
1. x > y
2. x - y = 12
Statements 1 and 2 together are not sufficient.
Each statement alone is sufficient.
Statement 2 alone is sufficient, but statement 1 alone is not sufficient 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.
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.
So, 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 #1 : Linear Equations, Two Unknowns
Given that 
Statement 1:
Statement 2:
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 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.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Solve for 
Statement 1:
Statement 2:
From either statement alone, it can be deduced that 
Example Question #7 : Dsq: Solving Linear Equations With Two Unknowns
Data Sufficiency Question
Solve for 
1.
2.
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
Statements 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
Each statement alone is sufficient
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 #121 : Algebra
Data Sufficiency Question
Solve for 
1.
2.
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
Each statement alone is sufficient
Statements 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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