Question

In the xy-plane, if line k has negative slope and passes through the point (-5,r), is the x-intercept of line k positive?
(1) The slope of line k is -5.
(2) r > 0

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For coordinate geometry problems, it can be helpful to visualize. The point \((-5, r)\) is in quadrant I if \(r>0\) or quadrant IV if \(r<0\). A line with a negative slope passing through a point in quadrant I could cross the x-axis on either the positive or negative side. The statements provide details that constrain the line, but you must check if the constraint is sufficient to guarantee only one outcome.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
We need to determine if the x-intercept of a line is positive. The x-intercept is the point where the line crosses the x-axis (i.e., where y=0).
We are given that the line has a negative slope (\(m<0\)) and passes through the point \((-5, r)\).
Step 2: Key Formula or Approach:
Using the point-slope form of a linear equation, \(y - y_1 = m(x - x_1)\), we have: \[ y - r = m(x - (-5)) \implies y - r = m(x + 5) \] To find the x-intercept, we set \(y = 0\): \[ 0 - r = m(x + 5) \] \[ -r = mx + 5m \] \[ -r - 5m = mx \] \[ x = \frac{-r - 5m}{m} = -\frac{r}{m} - 5 \] The question is: Is \(-\frac{r}{m} - 5>0\)?
Step 3: Detailed Explanation:
Analyze Statement (1): The slope of line k is -5 (\(m = -5\)).
Substituting \(m = -5\) into our question: Is \(-\frac{r}{-5} - 5>0\)?
Is \(\frac{r}{5} - 5>0\)?
Is \(\frac{r}{5}>5\)?
Is \(r>25\)?
We have no information about \(r\), so we cannot answer this. Statement (1) is not sufficient.
Analyze Statement (2): \(r>0\).
We know \(m<0\). The question is: Is \(-\frac{r}{m} - 5>0\)?
Since \(r>0\) and \(m<0\), the term \(\frac{r}{m}\) is negative, and \(-\frac{r}{m}\) is positive. Let's test some values.

Case 1 (Answer is "Yes"): Let \(r = 10\) and \(m = -1\). The conditions are met. The x-intercept is \(x = -\frac{10}{-1} - 5 = 10 - 5 = 5\). Since \(5>0\), the answer is "Yes".

Case 2 (Answer is "No"): Let \(r = 10\) and \(m = -3\). The conditions are met. The x-intercept is \(x = -\frac{10}{-3} - 5 = \frac{10}{3} - 5 = \frac{10 - 15}{3} = -\frac{5}{3}\). Since \(-\frac{5}{3}<0\), the answer is "No".

Since we can get both "Yes" and "No" answers, statement (2) is not sufficient.
Analyze Both Statements Together:
From statement (1), \(m = -5\). From statement (2), \(r>0\). The question simplifies to: Is \(r>25\)? We only know that \(r\) is a positive number. It could be \(r=10\) (in which case the answer is "No") or it could be \(r=30\) (in which case the answer is "Yes"). The information is still not sufficient.
Step 4: Final Answer:
Even with both statements, we cannot definitively determine if the x-intercept is positive.