To determine the value of [tex]\( r \)[/tex] such that the given pair of linear equations represent coincident lines, we need to ensure that the equations satisfy the condition for coincident lines. This condition states that the ratios of the corresponding coefficients must be equal.
Given the equations:
[tex]\[ 3x - y + 8 = 0 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 6x - ry + 16 = 0 \quad \text{(Equation 2)} \][/tex]
For the lines to be coincident:
[tex]\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \][/tex]
From the equations, we can extract the coefficients:
[tex]\[ a_1 = 3, \quad b_1 = -1, \quad c_1 = 8 \][/tex]
[tex]\[ a_2 = 6, \quad b_2 = -r, \quad c_2 = 16 \][/tex]
Now, let's equate the ratios:
[tex]\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2} \][/tex]
Next, we equate the constant terms:
[tex]\[ \frac{c_1}{c_2} = \frac{8}{16} = \frac{1}{2} \][/tex]
Since [tex]\(\frac{a_1}{a_2} = \frac{c_1}{c_2}\)[/tex], both ratios equal [tex]\( \frac{1}{2} \)[/tex]. For the lines to be coincident, the ratio of the [tex]\( y \)[/tex]-coefficients must also equal [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ \frac{b_1}{b_2} = \frac{-1}{-r} = \frac{1}{2} \][/tex]
Solving for [tex]\( r \)[/tex]:
[tex]\[ \frac{1}{r} = 2 \][/tex]
[tex]\[ r = \frac{1}{2} \][/tex]
Therefore, solving for [tex]\( r \)[/tex]:
[tex]\[ r = 2 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
