Parallel Lines - GMAT Quantitative

To find the equation of a parallel line, we need the slope and the y-intercept.

Parallel lines have the same slope, so we have that.

I and II each give us a point on the graph, so we could find the equation of the line through either of them.

Statement 1: We're given the slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.

Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.

We are asked to find the equation of a line related to another line.

Statement I tells us the two lines are parallel. This means they have the same slope

Statement II gives us a point on our desired line. We can use this to find the line's y-intercept, which will then allow us to write its equation.

Plug all of the given info into slope-intercept form and solve for b, the line's y-intercept:

So our equation is:

In order to calculate the equation of a line that passes through a point, we need one of two pieces of information. If we know another point, we can calculate the slope and solve for the -intercept, giving us the equation of the line. Alternatively, if we know the slope (which we can conclude from the parallel line in statement 2) we can calculate the -intercept and determine the equation of the line.

To find the equation of a parallel line, we need the slope and the y-intercept.

Parallel lines have the same slope, so we have that.

I and II each give us a point on the graph, so we could find the equation of the line through either of them.

Statement 1: We're given the slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.

Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.

We are asked to find the equation of a line related to another line.

Statement I tells us the two lines are parallel. This means they have the same slope

Statement II gives us a point on our desired line. We can use this to find the line's y-intercept, which will then allow us to write its equation.

Plug all of the given info into slope-intercept form and solve for b, the line's y-intercept:

So our equation is:

The lines are parallel, so their slopes are equal.

The slope of the first is .

The slope of the second is .

Set the two equal to each other:

If you know that , then you can easily find by substituting:

Cross-multiply and solve:

If you know that , do the same thing:

Therefore, either statement alone is sufficient to answer the question.

Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.

Assume both statements are true. Then three things are possible:

Case 1: Both lines will have slope 1, or

Case 2: Both lines will have slope

In either case, since the lines have the same slope, they are parallel.

Case 3: One line has slope 1 and one has slope

In this case the lines are perpendicular.

The two statements therefore provide insufficient information.

The question can be answered by finding and comparing the slopes of the lines. The lines are parallel if and only if they have the same slope, and perpendicular if and only if the product of the slopes is .

Statement 1 alone does not answer the question. Two lines with slope 1 are parallel, and a line with slope 2 and a line with slope are not, but in both cases, the product of the slopes is 1.

Statement 2 alone gives that Line has slope 1 or , but nothing is given about the slope of Line .

Now, assume both statements are true. From Statement 2, has slope 1 or . From Statement 1, the product of the slopes is 1; if the slope of is 1, then the slope of is , and if the slope of is , then the slope of is . Therefore, if both statements are true, the lines have the same slope, making them parallel.

The answer to the question depends on the slopes of the lines - parallel lines have the same slope, and perpendicular lines have slopes that have product .

Statement 1 alone only eliminates the possiblity of the lines being perpendicular, since the product of the slope is not . Two lines with slope 3 are parallel, and one line with slope 1 and one with slope 9 are neither parallel nor perpendicular; both pairs of lines satisfy Statement 1, but only the first pair is parallel. Therefore, Statement 1 only establishes that they are not perpendicular.

From Statement 2, only the slope of is given; without the slope of , the question cannot be answered.

Assume both statements to be true. Then since Line has slope and the product of the slopes is 9, The slope of Line is . Therefore, both lines have slope , and the lines are parallel.

The answer to the question depends on the slopes of the lines—parallel lines have the same slope, and perpendicular lines have slopes that have product .

From Statement 1 alone, we only know one point of each line, so no information about their slopes can be determined; the same holds for Statement 2.

Assume both statements are true. Then we know two points of each line—specifically, both intercepts—from which we can determine the slope of each by way of the slope formula. After doing so, we can use the slopes to answer the question.

Two lines are parallel if and only if they have the same slope, and perpendicular if and only if their slopes have product .

Assume Statement 1 alone. Since the product of the slopes is , the lines are perpendicular.

Statement 2 alone is unhelpful, since no information about the slope of a line can be determined from only one point.

Two lines can be determined to be parallel, perpendicular, or neither from their slopes.

Assume Statement 1 alone. Parallel lines must have the same slope, so this choice can be eliminated. The slopes of perpendicular lines must have product ; since the product of the slopes is , this choice can be eliminated as well. It can therefore be deduced that the lines are neither parallel nor perpendicular.

Assume Statement 2 alone. Since the lines have at least one point in common, they are not parallel, but this is the only choice that can be eliminated.

Two lines can be determined to be parallel, perpendicular, or neither from their slopes.

Assume Statement 1 alone is true. Since these distinct lines have the same slope, they are parallel.

Assume Statement 2 alone is true. No information about the slopes of the lines can be determined from one single point, so Statement 2 alone is insufficient.

The solution of a system of linear equations is the point at which their lines intersect; if they are parallel, then by definition, there is no such point, and the system has no solution.

If , then we can rewrite the second equation as

In slope-intercept form:

This line has a slope of . The other equation has a line with slope of also, as can be easily seen since it is already in slope-intercept form. Since both equations have lines with the same slope, they are either the same line or parallel lines; either way, the system does not have exactly one solution.

I) Tells us the two lines are parallel. Parallel lines have the same slope.

II) Gives us the x-intercept of b(t). By itself this gives us no clue as to the slop of b. If we had another point on b(t) we could find the slope, but we don't have another point.

So, statement I is what we need.

Statement 1: Since we're referring to a line parallel to line XY, the slopes will be identical. We can use the points provided to calculate the slope:

We can simplify the slope to just .

Statement 2: Finding the slope of a line parallel to line XY is really straightforward when given the equation of a line.

Where is the slope and the y-intercept.

In this case, our value is .\

Each statement alone is sufficient to answer the question.