Answer:
The correct answer is:
[tex]4 x -3 y = 16[/tex] and
[tex]8 x -6 y = 34[/tex] are the equations that have no solution.
Step-by-step explanation:
First of all, let us have a look at the rules for a pair of lines, that have solution or no solution.
Let the equations be:
[tex]a_1x+b_1y+c_1=0[/tex] and
[tex]a_2x+b_2y+c_2=0[/tex]
1. There exists exactly one solution:
[tex]\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}[/tex]
2. There exists infinitely many solutions i.e. lines are identical.
[tex]\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}[/tex]
3. There exists No solutions i.e. lines are parallel and will never intersect.
[tex]\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}[/tex]
Now, let us have a look at the given pair of lines:
Option a)
[tex]2 x + 8 y = 15\\4 x + 16 y = 30.[/tex]
The ratio:
[tex]\dfrac{2}{4}=\dfrac{8}{16}=\dfrac{15}{30} = \dfrac{1}{2}[/tex]
Hence, the lines are identical, infinite solutions.
Option b)
[tex]2 x - y = 18\\4 x + 2 y = 38[/tex]
The ratio:
[tex]\dfrac{2}{4}\neq \dfrac{-1}{2}[/tex]
Hence, exactly one solution.
Option c)
[tex]4 x + 7 y = 17\\8 x -14 y = 36[/tex]
The ratio:
[tex]\dfrac{4}{8}\neq \dfrac{-7}{14}\\\dfrac{1}{2}\neq \dfrac{-1}{2}[/tex]
Hence, exactly one solution.
Option d)
[tex]4 x - 3 y = 16\\8 x - 6 y = 34.[/tex]
The ratio:
[tex]\dfrac{4}{8}= \dfrac{-3}{-6}\neq\dfrac{16}{34}\\\Rightarrow \dfrac{1}{2}= \dfrac{1}{2}\neq\dfrac{8}{17}[/tex]
Hence, There is no solution.
The correct answer is:
[tex]4 x -3 y = 16[/tex] and
[tex]8 x -6 y = 34[/tex] are the equations that have no solution.
