Do the following equations represent a pair of coincidnet lines? Justify your answer.
`(x)/(2)+ y+(2)/(5)= 0`
`4x + 8y +(5)/(16)=0`

Answer

To determine whether the given equations represent a pair of coincident lines, we need to analyze the equations step by step.

The equations given are:
1. \(\frac{x}{2} + y + \frac{2}{5} = 0\)
2. \(4x + 8y + \frac{5}{16} = 0\)

### Step 1: Rewrite the first equation in standard form
To convert the first equation into standard form \(Ax + By + C = 0\), we can eliminate the fractions. We can multiply the entire equation by 10 (the least common multiple of the denominators 2 and 5) to clear the fractions:

\[ 10 \left(\frac{x}{2}\right) + 10y + 10\left(\frac{2}{5}\right) = 0 \]
This simplifies to:
\[ 5x + 10y + 4 = 0 \]

### Step 2: Identify coefficients of the first equation
From the first equation \(5x + 10y + 4 = 0\), we can identify:
- \(A_1 = 5\)
- \(B_1 = 10\)
- \(C_1 = 4\)

### Step 3: Rewrite the second equation in standard form
Now, let's rewrite the second equation \(4x + 8y + \frac{5}{16} = 0\). We can eliminate the fraction by multiplying the entire equation by 16:

\[ 16(4x) + 16(8y) + 16\left(\frac{5}{16}\right) = 0 \]
This simplifies to:
\[ 64x + 128y + 5 = 0 \]

### Step 4: Identify coefficients of the second equation
From the second equation \(64x + 128y + 5 = 0\), we can identify:
- \(A_2 = 64\)
- \(B_2 = 128\)
- \(C_2 = 5\)

### Step 5: Check for coincident lines
For the lines to be coincident, the ratios of the coefficients must be equal:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

Calculating the ratios:
\[ \frac{A_1}{A_2} = \frac{5}{64}, \quad \frac{B_1}{B_2} = \frac{10}{128}, \quad \frac{C_1}{C_2} = \frac{4}{5} \]

### Step 6: Simplify the ratios
Now, we simplify the second ratio:
\[ \frac{10}{128} = \frac{5}{64} \quad (\text{Dividing both numerator and denominator by 2}) \]

Now we have:
- \(\frac{A_1}{A_2} = \frac{5}{64}\)
- \(\frac{B_1}{B_2} = \frac{5}{64}\)
- \(\frac{C_1}{C_2} = \frac{4}{5}\)

### Step 7: Compare the ratios
Since \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\) but \(\frac{C_1}{C_2}\) is not equal to these, we conclude that the lines are not coincident.

### Conclusion
The given equations do not represent a pair of coincident lines. They are parallel lines since the ratios of the coefficients \(A\) and \(B\) are equal, but the ratio of \(C\) is different.

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