Number of solution in the given pair of equation is infinitely many solutions
`x + 2y - 8 = 0`
` 2x + 4y = 16`

Answer

To determine if the given pair of equations has infinitely many solutions, we can follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \( x + 2y - 8 = 0 \)
2. \( 2x + 4y = 16 \)

We can rewrite the second equation in standard form:
\[ 2x + 4y - 16 = 0 \]

### Step 2: Identify coefficients
Now, we can identify the coefficients from both equations:
- For the first equation \( x + 2y - 8 = 0 \):
- \( a_1 = 1 \)
- \( b_1 = 2 \)
- \( c_1 = -8 \)

- For the second equation \( 2x + 4y - 16 = 0 \):
- \( a_2 = 2 \)
- \( b_2 = 4 \)
- \( c_2 = -16 \)

### Step 3: Check the ratios
To check if the equations have infinitely many solutions, we need to verify the condition:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Calculating the ratios:
1. \( \frac{a_1}{a_2} = \frac{1}{2} \)
2. \( \frac{b_1}{b_2} = \frac{2}{4} = \frac{1}{2} \)
3. \( \frac{c_1}{c_2} = \frac{-8}{-16} = \frac{1}{2} \)

### Step 4: Compare the ratios
Since all three ratios are equal:
\[ \frac{1}{2} = \frac{1}{2} = \frac{1}{2} \]

### Conclusion
Since the condition is satisfied, we conclude that the given pair of equations has infinitely many solutions.

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