Does the Line with Equation ax + by = c, Where a, b and c are Real Constants GMAT Data Sufficiency

Question: Does the line with equation ax + by = c, where a, b and c are real constants, cross the x-axis?

(1) b ≠ 0
(2) ab > 0

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: B
Solution and Explanation:
Approach Solution 1:

Given in the question that the line has an equation ax +by = c. It has asked to find out if it crosses the x-axis.
Line ax+by=c will have a point at (z, 0) if it crosses the x-axis. which implies:
Z= (c/a) or az = c. When an is not equal to zero, it will be possible for z to have a real value.
Thus, the question is: Is it equal to 0?

assertion 1: b is not equal to 0; nothing about an is revealed by this. NS
assertion 2: an is unquestionably not equal to 0, hence ab > 0. Sufficient.

B is the solution.
Note: The result is (b)(y)=0 if we use the equation z = (c/a) knowing that an is not equal to zero. Y must be zero because assertion (2) also asserts that b is not equal to zero. This is what we want and demonstrates how this line cuts the x axis.

Approach Solution 2:

Given the question that the line has an equation ax +by = c. It has asked to find out if it crosses the x-axis.

We are asked to determine whether a line with the equation ax+by = c, where a, b, and c are real constants, crosses the x-axis. Let's begin by focusing on y alone in the above equation:

ax+by = c
by = -ax + c
(-a/b)x + (c/b) = y

The slope is now -a/b and the y-intercept is c/b, giving us the provided equation in slope-intercept form.

One Statement Only:
not equivalent to 0 in b

Statement 1 is insufficient because we are not given any information regarding the slope of the line. Answer options A and D should be dropped.

Only Statement Two:

ab > 0
Both a and b cannot be zero since ab > 0. This indicates that the line's slope is not zero. Remember that a line is horizontal if its slope equals 0, and it won't cross the x-axis unless it is the x-axis itself. Hence, the line will eventually cross the x-axis regardless of the values of an or b.

B is the solution

Approach Solution 3:

Given the question that the line has an equation ax +by = c. It has asked to find out if it crosses the x-axis.

If a line's slope is not zero, it will always cross the x-axis.
You can write axe + by = c as y=(ab)x+ (cb)
Putting it up against y=mx+c
where the slope is m
that is, SLope of the specified line = ab
REPHRASED: Is ab=0 true?

First claim: b 0.

however because a could be 0 or not,
NON EXTENSIVE
Stipulation 2: ab > 0

i.e. ab≠0

Since the slope is non-zero and always crosses the x-axis
SUFFICIENT

B is the solution

“Does the line with equation ax + by = c, where a, b and c are real constants”- is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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