Linear Equations, Two Unknowns - GMAT Quantitative

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.

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.

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

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.

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.

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.

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.

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.

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.

Solve for in each statement.

Statement 1:

Statement 2:

From either statement alone, it can be deduced that .

Let:

t: the number of trucks sold last week

d: the number of dolls sold last week - d is the value we are looking to find

To evaluate the statements, we translate the word problems into equations

(1): or

(2):

Each statement provides a single equation with two unknowns which is unsolvable, so each statement alone is not enough.

The two statements taken together give us a system of two equations with two unknowns, which we can solve.

Therefore the right answer is C.

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.

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.

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

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.

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.

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.

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.

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.

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.