Question

Find the equation of a line.
1. Its x and y intercept is 2 and -2 respectively.
2. The slope of the line is 1.

• 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 to answer the question ask
• 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 specific to the problem are needed.

Hint:

Knowing the slope of a line only tells you its steepness or orientation. To pin down its exact location, you also need at least one point that the line passes through (like the y-intercept). Statement (1) provides two points, which is more than enough information.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To uniquely determine the equation of a line, we need information that fixes its position and orientation on the coordinate plane. This can be achieved by knowing:
1. Two distinct points on the line.
2. One point on the line and its slope.
3. The slope and the y-intercept.

Step 2: Key Formula or Approach:
The slope-intercept form of a linear equation is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
The intercept form is \( \frac{x}{a} + \frac{y}{b} = 1 \), where \( a \) is the x-intercept and \( b \) is the y-intercept.

Step 3: Detailed Explanation:
Analyzing Statement (1): Its x and y intercept is 2 and -2 respectively.
This gives us two distinct points on the line:
- The x-intercept is 2, which corresponds to the point (2, 0).
- The y-intercept is -2, which corresponds to the point (0, -2).
Since we have two points, we can uniquely determine the line. We can find the slope:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 0}{0 - 2} = \frac{-2}{-2} = 1 \] The y-intercept (\( c \)) is given as -2. So, the equation is \( y = 1x - 2 \) or \( y = x - 2 \).
Since a unique equation can be found, statement (1) is sufficient.
Analyzing Statement (2): The slope of the line is 1.
This tells us that \( m = 1 \). The equation of the line is of the form \( y = x + c \).
However, the y-intercept \( c \) is unknown. There are infinitely many lines with a slope of 1 (e.g., \( y = x+1, y = x+5, y = x-10 \)). These are all parallel lines.
Since we cannot find a unique equation, statement (2) is not sufficient.

Step 4: Final Answer:
Statement (1) alone is sufficient, but statement (2) alone is not.