Question

The graph of which of the following equations is a straight line that is parallel to line M in the figure above and intersects the negative direction of Y-axis?
Indicate all such equations.
[Note: Select one or more answer choices]
\includegraphics[width=0.7\linewidth]{04.png}

Hint:

For equations in the standard form \(Ax + By = C\), the slope is always \(-\frac{A}{B}\) and the y-intercept is \(\frac{C}{B}\). To find parallel lines to \(4y-3x=\text{const}\) (or \( -3x+4y=\text{const} \)), you just need to find equations of the form \(-3x+4y=C'\) or \(3x-4y=C''\). The slope will be \(-(-3)/4 = 3/4\). Then check the y-intercept condition. A negative y-intercept \(\frac{C'}{4}\) means \(C'\) must be negative.

Answer 1

Step 1: Understanding the Concept:
We need to find equations of lines that satisfy two conditions: 1. Parallel to line M: Two lines are parallel if and only if they have the same slope. 2. Intersects the negative direction of Y-axis: This means the y-intercept of the line must be negative. First, we'll find the slope of line M from the given points. Then, we'll find the slope and y-intercept of each equation in the options to see which ones meet the criteria.
Step 2: Key Formula or Approach:
1. Calculate the slope of line M using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). The given points are (-4, 0) and (0, 3). 2. Convert each of the given equations into the slope-intercept form, \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. 3. Compare the slope of each option with the slope of line M. They must be equal. 4. For the equations with the correct slope, check if their y-intercept \(c\) is negative (\(c<0\)).
Step 3: Detailed Explanation:
Find the slope of line M:
Line M passes through the points \((-4, 0)\) and \((0, 3)\). \[ m_M = \frac{3 - 0}{0 - (-4)} = \frac{3}{4} \] So, we are looking for lines with a slope of \(\frac{3}{4}\) and a negative y-intercept.
Analyze the options:
We will convert each equation to the form \(y = mx + c\).
% Option A: 4y+3x=0
\(4y = -3x \implies y = -\frac{3}{4}x\). Slope is \(-\frac{3}{4}\). Not parallel.
% Option B: 4y-3x=-2
\(4y = 3x - 2 \implies y = \frac{3}{4}x - \frac{2}{4} = \frac{3}{4}x - \frac{1}{2}\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = -\frac{1}{2}\) (negative). This is a correct choice.
% Option C: 4y-3x=4
\(4y = 3x + 4 \implies y = \frac{3}{4}x + 1\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = 1\) (positive). Incorrect.
% Option D: 4y+3x=-4
\(4y = -3x - 4 \implies y = -\frac{3}{4}x - 1\). Slope is \(-\frac{3}{4}\). Not parallel.
% Option E: 4y-3x=-1
\(4y = 3x - 1 \implies y = \frac{3}{4}x - \frac{1}{4}\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = -\frac{1}{4}\) (negative). This is a correct choice.
% Option F: 4y-3x=0
\(4y = 3x \implies y = \frac{3}{4}x\). Slope is \(m = \frac{3}{4}\) (parallel). Y-intercept is \(c = 0\) (not negative). Incorrect.
Step 4: Final Answer:
The equations that represent a line parallel to M and intersect the negative y-axis are 4y-3x=-2 and 4y-3x=-1.