DSQ: Calculating x or y intercept - GMAT Quantitative

To find the - and y-intercepets, we need both statements of information.

Statement 1 is not enough information because we don't have a reference point for the slope. A slope by itself is not enough to define the line. There are an infinite amount of lines with a slope of 0.6.

Statement 2 is not enough information by itself since it only tells us about 1 point on the line. Again, there are an infinite number of lines that pass through the point (10,2).

Only by using both statements can we find the - and y-intercepts. Solving, we see the line is actually

The -intercept of the graph of an equation is the point at which , so we evaluate :

The -intercept is simply the point , so knowing is necessary, and knowing is neither necessary nor helpful.

To find the -intercept of , evaluate :

Knowing is necessary and sufficient; the value of is irrelevant.

To find the -intercept of , evaluate :

Knowing both and is necessary and sufficient.

Two points are necessary and sufficient to define a line. Therefore, neither statement alone is sufficient to determine the line, but both are sufficient. Once the line is defined, the -intercept - the point at which the line intersects the -axis - can be determined.

The -intercept of the graph of is the point at which it intersects the -axis. Since this point has -coordinate 0, the -coordinate is . Statement 1 does not give us this value, but Statement 2 does.

The two statements together prvide insufficient information.

Assume both statements are true. By Statement 1, is a constantly increasing function, so it can intersect the -axis at most one time.

Now examine these two cases.

Case 1:

.

Also, if , then , so .

Since , the function has exactly one -intercept.

Case 2:

Also, if , then , so .

However, 2 raised to any power must be positive, so there is no value for which . The function has no -intercepts.

Statement 1 alone establishes that is always increasing. Its graph cannot have more than one -intercept; if it does, then the graph of the function must have a vertex between two intercepts, violating this statement. But it does not answer the question as to how many intercepts has, as seen in these two cases:

Case 1:

This is a linear function that is always increasing—it is in slope-intercept form, and its slope is 1, a positive number. The graph of has exactly one -intercept.

Case 2:

An exponential function with a base greater than 1, such as this, is an increasing function; however, 2 raised to any power must be positive, so there is no value for which . The graph of has no -intercepts.

Statement 2 alone establishes that at least one -intercept exists - since , is an -intercept. It does not, however, rule out the possibility of more -intercepts.

Now assume both statements are true. Since Statement 1 establishes that there is at most one -intercept, and Statement establishes that there is at least one -intercept, the two statements together establish that there is exactly one.

Assume both statements are true.

By Statement 1, is decreasing on the domain interval ; by Statement 2, is increasing on the domain interval . Therefore, must have its minimum value when .

This does not, however, tell us the number of -intercepts. For example, the graph of has as its minimum point , and, subsequently, exactly one -intercept. The graph of has as its minimum point and, subsequently, no -intercepts.

The -intercept of the graph of is the point at which it intersects the -axis. Since this point has -coordinate 0, the -coordinate is the value for which . Statement 1 gives us this value; Statement 2 does not.

Statement 1 provides insufficient information, since the slope of the line alone is not enough from which to deduce the -intercept. Statement 2 alone tells us that the line passes through the point ; since this is on the -axis, this is the -intercept.

The question is equivalent to asking whether the lines intersect at a point on the -axis.

Assume Statement 1 alone. Since and are distinct lines, , their common -intercept, is their sole point of intersection. They cannot intersect at a second point, so they cannot have the same -intercept.

Assume Statement 2 alone. Perpendicular lines are lines that meet at right angles; the question of their point of intersection is not answered by this statement.

The question is equivalent to asking whether the lines intersect at a point on the -axis.

Statement 1 only establishes that the lines pass through different points on the -axis; no clues are given about any of their other points.

Statement 2 establishes that the lines have the same slope, and, subsequently, are parallel - that is, they do not intersect at all. Therefore, they cannot have the same -intercept.

Let stand for the values Stanley wrote in the square, the circle, and the triangle, respectively. The equation becomes

.

To find the -coordinate of the -intercept, set and solve for :

It is therefore necessary and sufficient to know the values of and - the values Ralph wrote in the square and the triangle - in order to determine the -intercept of Ralph's equation. Each statement alone give only one of those numbers; the two together give both.

Let stand for the values Ralph wrote in the square, the circle, and the triangle, respectively. The equation becomes

.

To find the -coordinate of the -intercept, set and solve for :

It is therefore necessary and sufficient to know the values of and - the values Ralph wrote in the circle and the triangle - in order to determine the -intercept of Ralph's equation. Statement 1 is irrelevant, and Statement 2 only provides the value Ralph wrote in the triangle.

If we let stand for the number Gloria wrote in the square and stand for the number she wrote in the circle, the equation becomes

,

the slope-intercept form of the equation of a line. We can find the -coordinate of the -intercept by setting and solving for :

Therefore, it is necessary and sufficient to know both and - both of the numbers Gloria wrote - to determine the -intercept of the equation she made. Neither statement provides both values, but both statements together do.

If we let stand for the number Ray wrote in the square and stand for the number he wrote in the circle, the equation becomes

,

the slope-intercept form of the equation of a line. In this form, the -intercept is solely determined by the value of —namely, the value that Ray wrote in the circle. Statement 1 is therefore irrelevant, and Statement 2 alone establishes that Ray did not succeed.

Let stand for the values Mandy wrote in the square, the circle, and the triangle, respectively. The equation becomes

.

From Statement 1 alone, we know that Mandy wrote a 5 in the triangle, but we do not know any of the others. The question of Mandy's success is unresolved.

Now assume Statement 2 alone is known. Then

.

The equation can be rewritten as

and it can be rewritten in slope-intercept form as

Mandy wrote an equation whose line has slope .

However, the slope of a line through and is

.

Statement 2 alone is sufficient to establish that Mandy did not succeed.

The -intercept is the point at which . To find the -coordinate of the -intercept, just substitute 0 for :

Therefore, you need only know ; knowing is neither necessary nor helpful.

If you are given that since . the -intercept is easily determined to be .

The answer is that Statement 2 alone is sufficient, but not Statement 1 alone.