Check whether the graph of the pair of linear equations `x + 2y – 4 = 0` and `2x + 4y – 12 = 0` is intersecting lines or parallel lines.

Answer

To determine whether the graphs of the given pair of linear equations are intersecting lines or parallel lines, we can follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( x + 2y - 4 = 0 \) (Equation 1)
2. \( 2x + 4y - 12 = 0 \) (Equation 2)

### Step 2: Rearrange the equations to slope-intercept form
We can rearrange both equations into the slope-intercept form \( y = mx + b \).

**For Equation 1:**
\[ x + 2y - 4 = 0 \implies 2y = -x + 4 \implies y = -\frac{1}{2}x + 2 \]
Here, the slope \( m_1 = -\frac{1}{2} \).

**For Equation 2:**
\[ 2x + 4y - 12 = 0 \implies 4y = -2x + 12 \implies y = -\frac{1}{2}x + 3 \]
Here, the slope \( m_2 = -\frac{1}{2} \).

### Step 3: Compare the slopes
Both equations have the same slope:
\[ m_1 = m_2 = -\frac{1}{2} \]

### Step 4: Check the y-intercepts
The y-intercepts are:
- For Equation 1: \( b_1 = 2 \)
- For Equation 2: \( b_2 = 3 \)

Since the slopes are equal but the y-intercepts are different, this indicates that the lines are parallel.

### Conclusion
The graphs of the equations \( x + 2y - 4 = 0 \) and \( 2x + 4y - 12 = 0 \) are parallel lines.

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