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.
• 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

Hint:

For linear equations, use the point-slope form to derive the equation and solve for the \( x \)-intercept.

Answer 1

The Correct Option is A

Step 1: Analyze statement (1).
Statement (1) tells us that the slope of the line is \( -5 \), and the line passes through the point \((-5, r)\). We can write the equation of the line in point-slope form: \[ y - r = -5(x + 5) \] Simplifying: \[ y = -5x - 25 + r \] The \( x \)-intercept occurs when \( y = 0 \): \[ 0 = -5x - 25 + r \quad \implies \quad 5x = -25 + r \quad \implies \quad x = \frac{-25 + r}{5} \] For the \( x \)-intercept to be positive, \( -25 + r \) must be positive, meaning \( r>25 \). Thus, statement (1) alone is sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that \( r>0 \), but this does not provide any information about the \( x \)-intercept or the slope, so statement (2) alone is not sufficient.
\[ \boxed{A} \]