The lines with the equations `y = m_(1)x+4` and `y = m_(2)x+3` will intersect to the right of the y - axis if and only is

Options

A. `m_(1)=m_(2)`
B. `m_(1)lt m_(2)`
C. `m_(1)gt m_(2)`
D. `m_(1)+m_(2)=0`

Explanation

To determine the condition under which the lines represented by the equations \(y = m_1 x + 4\) and \(y = m_2 x + 3\) intersect to the right of the y-axis, we will follow these steps:

### Step 1: Set the equations equal to find the intersection point
We start by equating the two equations to find the x-coordinate of the intersection point:
\[ m_1 x + 4 = m_2 x + 3 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to isolate the terms involving \(x\):
\[ m_1 x - m_2 x = 3 - 4 \]
This simplifies to:
\[ (m_1 - m_2)x = -1 \]

### Step 3: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{-1}{m_1 - m_2} \]

### Step 4: Determine the condition for \(x\) to be positive
For the intersection point to be to the right of the y-axis, \(x\) must be positive:
\[ \frac{-1}{m_1 - m_2} > 0 \]
This inequality implies that \(m_1 - m_2\) must be negative (since a negative divided by a negative is positive). Therefore:
\[ m_1 - m_2 < 0 \quad \Rightarrow \quad m_1 < m_2 \]

### Conclusion
Thus, the condition for the lines to intersect to the right of the y-axis is:
\[ m_1 < m_2 \]