GMAT Math : DSQ: Solving linear equations with two unknowns

Study concepts, example questions & explanations for GMAT Math

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Example Question #1 : Linear Equations, Two Unknowns

What is the value of z?

Statement 1: \dpi{100} \small x+y+z=4

Statement 2: \dpi{100} \small 2x+y^{2}+z=17

Possible Answers:

EACH statement ALONE is sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Correct answer:

Statements 1 and 2 TOGETHER are NOT sufficient.

Explanation:

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 #1 : Dsq: Solving Linear Equations With Two Unknowns

Is the equation linear?

Statement 1:

Statement 2:  is a constant

Possible Answers:

Statement 1 ALONE is sufficient, but statement 2 is not 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.

EACH statement ALONE is sufficient.

Correct answer:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

If we only look at statement 1, we might think the equation is not linear because of the  term.  But statement 2 tells us the  is a constant.  Then the equation is linear.  We need both statements to answer this question.

Example Question #1 : Linear Equations, Two Unknowns

Data sufficiency question- do not actually solve the question

Solve for :

\small 2x+3xy+4y=7

1. \small x=1

2. \small x+4y=13

Possible Answers:

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

Each statement alone is sufficient

Both statements taken together are sufficient to answer the question, but neither question alone is sufficient

Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question

Correct answer:

Each statement alone is sufficient

Explanation:

When solving an equation with 2 variables, a second equation or the solution of 1 variable is necessary to solve.

Example Question #4 : Linear Equations, Two Unknowns

Data Sufficiency Question

Solve for  and .

1.

2. Both  and  are positive integers

Possible Answers:

both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

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

each statement alone is sufficient

Correct answer:

each statement alone is sufficient

Explanation:

Using statement 1 we can set up a series of equations and solve for both  and .

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 #2 : Linear Equations, Two Unknowns

Given that both  , how many solutions does this system of equations have: one, none, or infinitely many?

Statement 1: 

Statement 2: 

Possible Answers:

Statement 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.

Statement 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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

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 , meaning that the lines have unequal slopes, and that there is only one solution. Statement 2 tells us the value of , which is irrelevant.

Example Question #1 : 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: 

Possible Answers:

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.

EITHER 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.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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  and  in order to determine their equality or inequality, and only if they are unequal can we answer the question.

Set  and .

The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.

Example Question #2 : Dsq: Solving Linear Equations With Two Unknowns

Solve the following for x:

4x+7y = 169

1. x > y

2. x - y = 12

 

Possible Answers:

Statements 1 and 2 together are not sufficient.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient 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.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Correct answer:

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

Explanation:

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 #8 : Linear Equations, Two Unknowns

Given that , evaluate .

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Solve for  in each statement.

Statement 1:

Statement 2:

From either statement alone, it can be deduced that .

 

Example Question #3 : Linear Equations, Two Unknowns

Data Sufficiency Question

Solve for  and :

 

1.

2.

Possible Answers:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Each statement alone is sufficient

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

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

Correct answer:

Each statement alone is sufficient

Explanation:

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 : Linear Equations, Two Unknowns

Data Sufficiency Question

Solve for , , and :

 

1.

2.

Possible Answers:

Statement 1 alone is sufficient, but statement 2 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

Each statement alone is sufficient

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Explanation:

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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