The two lines 4 x + 3y =0 and 7x + 5y = 0 will _______ in their graphical representation

Options

A. be parallel to each other
B. intersect each other at one point only
C. intersect eah other at three points only
D. coincide each other

Answer

To determine how the two lines \(4x + 3y = 0\) and \(7x + 5y = 0\) will behave in their graphical representation, we need to analyze their slopes and whether they intersect.

### Step 1: Convert the equations to slope-intercept form
To find the slopes of the lines, we can rewrite both equations in the form \(y = mx + b\), where \(m\) is the slope.

**For the first line \(4x + 3y = 0\):**
1. Rearrange the equation to isolate \(y\):
\[ 3y = -4x \]
2. Divide by 3:
\[ y = -\frac{4}{3}x \]
Here, the slope \(m_1 = -\frac{4}{3}\).

**For the second line \(7x + 5y = 0\):**
1. Rearrange the equation to isolate \(y\):
\[ 5y = -7x \]
2. Divide by 5:
\[ y = -\frac{7}{5}x \]
Here, the slope \(m_2 = -\frac{7}{5}\).

### Step 2: Compare the slopes
Now we compare the slopes of the two lines:
- Slope of the first line \(m_1 = -\frac{4}{3}\)
- Slope of the second line \(m_2 = -\frac{7}{5}\)

Since \(m_1 \neq m_2\), the lines have different slopes.

### Step 3: Conclusion about the lines
Since the slopes of the two lines are different, this means that the lines are not parallel and will intersect at exactly one point.

### Final Answer
The two lines \(4x + 3y = 0\) and \(7x + 5y = 0\) will intersect at one point in their graphical representation.

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