Question

In a rectangular coordinate system, line k has x-intercept 4 and slope –2.
Column A: The y-intercept of k
Column B: 2

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

An intercept is a point. An \(x\)-intercept of \(c\) means the point \((c, 0)\) is on the line. A \(y\)-intercept of \(b\) means the point \((0, b)\) is on the line. You can use the given point and slope to write the equation of the line and then find the desired intercept.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the \(y\)-intercept of a line, given its \(x\)-intercept and slope. This involves using the concepts of linear equations.
Step 2: Key Formula or Approach:
The equation of a line can be written in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.
Alternatively, we can use the point-slope form, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line.
Step 3: Detailed Explanation:
Column A: We need to find the \(y\)-intercept of line \(k\).
We are given the following information:

Slope (\(m\)) = -2
\(x\)-intercept = 4. The \(x\)-intercept is the point where the line crosses the x-axis, which means the y-coordinate is 0. So, the point \((4, 0)\) is on the line.
Using the point-slope form \(y - y_1 = m(x - x_1)\):
\[ y - 0 = -2(x - 4) \] \[ y = -2x + 8 \] This is now in the slope-intercept form \(y = mx + b\). By comparing the two, we can see that the \(y\)-intercept, \(b\), is 8.
So, the quantity in Column A is 8.
Column B: The quantity is given as 2.
Comparison:
We compare 8 (Column A) and 2 (Column B).
Since \(8>2\), the quantity in Column A is greater.
Step 4: Final Answer:
The \(y\)-intercept of the line is 8, which is greater than 2.